# BigNum Bakeoff Reboot

Some of you may be familiar with the BigNum Bakeoff, which ended up quite interestingly. The goal can more or less be summarized as writing a C program who's output would be the largest, under some constraints and theoretical conditions e.g. a computer that could run the program.

In the same spirit, I'm posing a similar challenge open to all languages. The conditions are:

• Maximum 512 bytes.

• Final result must be printed to STDOUT. This is your score. If multiple integers are printed, they will be concatenated.

• Output must be an integer. (Note: Infinity is not an integer.)

• No built-in constants larger than 10, but numerals/digits are fine (e.g. Avogadro's constant (as a built-in constant) is invalid, but 10000 isn't.)

• The program must terminate when provided sufficient resources to be run.

• The printed output must be deterministic when provided sufficient resources to be run.

• You are provided large enough integers or bigints for your program to run. For example, if your program requires applying basic operations to numbers smaller than 101,000,000, then you may assume the computer running this can handle numbers at least up to 101,000,000. (Note: Your program may also be run on a computer that handles numbers up to 102,000,000, so simply calling on the maximum integer the computer can handle will not result in deterministic results.)

• You are provided enough computing power for your program to finish executing in under 5 seconds. (So don't worry if your program has been running for an hour on your computer and isn't going to finish anytime soon.)

• No external resources, so don't think about importing that Ackermann function unless it's a built-in.

All magical items are being temporarily borrowed from a generous deity.

## Extremely large with unknown limit

where B³F is the Church-Kleene ordinal with the fundamental sequence of

B³F[n] = B³F(n), the Busy Beaver BrainF*** variant
B³F[x] = x, ω ≤ x < B³F


1. Simply Beautiful Art, Ruby fψ0(X(ΩM+X(ΩM+1ΩM+1)))+29(999)

2. Steven H, Pyth fψ(ΩΩ)+ω²+183(25627!)

3. Leaky Nun, Python 3 fε0(999)

4. fejfo, Python 3 fωω6(fωω5(9e999))

5. Steven H, Python 3 fωω+ω²(99999)

6. Simply Beautiful Art, Ruby fω+35(9999)

7. i.., Python 2, f3(f3(141))

## Some side notes:

If we can't verify your score, we can't put it on the leaderboard. So you may want to expect explaining your program a bit.

Likewise, if you don't understand how large your number is, explain your program and we'll try to work it out.

If you use a Loader's number type of program, I'll place you in a separate category called "Extremely large with unknown limit", since Loader's number doesn't have a non-trivial upper bound in terms of the fast growing hierarchy for 'standard' fundamental sequences.

Numbers will be ranked via the fast-growing hierarchy.

For those who would like to learn how to use the fast growing hierarchy to approximate really large numbers, I'm hosting a Discord server just for that. There's also a chat room: Ordinality.

Similar challenges:

Largest Number Printable

Golf a number bigger than TREE(3)

Shortest terminating program whose output size exceeds Graham's number

For those who want to see some simple programs that output the fast growing hierarchy for small values, here they are:

### Ruby: fast growing hierarchy

#f_0:
f=->n{n+=1}

#f_1:
f=->n{n.times{n+=1};n}

#f_2:
f=->n{n.times{n.times{n+=1}};n}

#f_3:
f=->n{n.times{n.times{n.times{n+=1}}};n}

#f_ω:
f=->n{eval("n.times{"*n+"n+=1"+"}"*n);n}

#f_(ω+1):
f=->n{n.times{eval("n.times{"*n+"n+=1"+"}"*n)};n}

#f_(ω+2):
f=->n{n.times{n.times{eval("n.times{"*n+"n+=1"+"}"*n)}};n}

#f_(ω+3):
f=->n{n.times{n.times{n.times{eval("n.times{"*n+"n+=1"+"}"*n)}}};n}

#f_(ω∙2) = f_(ω+ω):
f=->n{eval("n.times{"*n+"eval(\"n.times{\"*n+\"n+=1\"+\"}\"*n)"+"}"*n);n}


etc.

To go from f_x to f_(x+1), we add one loop of the n.times{...}.

Otherwise, we're diagonalizing against all the previous e.g.

f_ω(1) = f_1(1)
f_ω(2) = f_2(2)
f_ω(3) = f_3(3)

f_(ω+ω)(1) = f_(ω+1)(1)
f_(ω+ω)(2) = f_(ω+2)(2)
f_(ω+ω)(3) = f_(ω+3)(3)


etc.

• Do numerals count as built-in constants? – PyRulez Nov 24 '17 at 22:34
• @CloseVoters How can this be too broad... Well, asking the user to output one number in infinitely many numbers is not the same as asking the user to choose one of infinitely many tasks to do. To be fair this question ask the user to do the same thing too. 4 close votes as too broad already... – user202729 Nov 26 '17 at 2:13
• @Οurous Yes, you may assume that. But realize that when your program is given more resources, including faster computation, the output must still be deterministic. – Simply Beautiful Art Dec 9 '17 at 1:40
• I stated in the other comment section why I think the bounded Brainfuck Busy Beaver function will be exponential, but I'd like to add that more generally, I don't think the Church-Kleene ordinal will be the appropriate level for any computer program. A function one can code with a program are computable, and so should fall into the provably recursive functions of some sufficiently strong recursive sound theory. That theory will have a recursive proof theoretic ordinal, and that function will be below that ordinal in the FGH, assuming reasonable fundamental sequences. – Deedlit Dec 13 '17 at 6:28
• Of course the actual Busy Beaver function cannot be coded into program (hypercomputational languages aside), and restricted Busy Beaver functions that can be programmed must by necessity be much slower growing. – Deedlit Dec 13 '17 at 6:30

# Ruby, fψ0(X(ΩM+X(ΩM+1ΩM+1)))+29(999)

where M is the first Mahlo 'ordinal', X is the chi function (Mahlo collapsing function), and ψ is the ordinal collapsing function.

f=->a,n,b=a,q=n{c,d,e=a;!c ?[q]:a==c ?a-1:e==0||e&&d==0?c:e ?[[c,d,f[e,n,b,q]],f[d,n,b,q],c]:n<1?9:!d ?[f[b,n-1],c]:c==0?n:[t=[f[c,n],d],n,c==-1?[]:d==0?n:[f[d,n,b,t]]]};(x=9**9**9).times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{x.times{h=[];x.times{h=[h,h,h]};h=[[-1,1,[h]]];h=f[h,p x*=x]until h!=0}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}


Try it online!

### Code Breakdown:

f=->a,n,b=a,q=n{          # Declare function
c,d,e=a;          # If a is an integer, c=a and d,e=nil. If a is an array, a=[c,d,e].compact, and c,d,e will become nil if there aren't enough elements in a (e.g. a= #=> c=1,d=e=nil).
!c ?[q]:          # If c is nil, return [q], else
a==c ?a-1:          # If a==c, return a-1, else
e==0||e&&d==0?c:          # If e==0 or e is not nil and d==0, return c, else
e ?[[c,d,f[e,n,b,q]],f[d,n,b,q],c]:          # If e is not nil, return an array inside an array, else
n<1?9:          # If n<1, return 9, else
!d ?[f[b,n-1],c]:          # If d is nil, return [f[b,n-1],c], else
c==0?n:          # If c==0, return n, else
[t=[f[c,n],d],n,c==-1?[]:d==0?n:[f[d,n,b,t]]]          # t=[f[c,n],d]. If c==-1, return [t,n,[]], else if d==0, return [t,n,n], else return [t,n,[f[d,n,b,t]]].
};          # End of function
(x=9**9**9)          # Declare x
x.times{...}          # Looped within 33 x.times{...} loops
h=[];          # Declare h
x.times{h=[h,h,h]};          # Nest h=[h,h,h] x times
h=f[h,p x*=x]          # Apply x*=x, print x, then h=f[h,x]
until h==0          # Repeat previous line until h==0


### Math Breakdown:

f reduces a based on n,b,q.

The basic idea is to have an extremely nested a and reduce it repeatedly until it reduces down to a=0. For simplicity, let

g[0,n]=n
g[a,n]=g[f[a,n],n+1]


For now, let's only worry about n.

For any integer k, we get f[k,n]=k-1, so we can see that

g[k,n]=n+k


We then have, for any d, f[[0,d],n]=n, so we can see that

g[[0,d],n]
= g[f[[0,d],n],n+1]
= g[n,n+1]
= n+n+1


We then have, for any c,d,e, f[[c,0,e],n]=f[[c,d,0],n]=c. For example,

g[[[0,d],0,e],n]
= g[f[[[0,d],0,e]],n+1]
= g[[0,d],n+1]
= (n+1)+(n+1)+1
= 2n+3


We then have, for any c,d,e such that it does not fall into the previous case, f[[c,d,e],n]=[[c,d,f[e,n]],f[d,n],e]. This is where it starts to get complicated. A few examples:

g[[[0,d],1,1],n]
= g[f[[[0,d],1,1],n],n+1]
= g[[[0,d],1,0],0,[0,d]],n+1]
= g[f[[[0,d],1,0],0,[0,d]],n+1],n+2]
= g[[[0,d],1,0],n+2]
= g[f[[[0,d],1,0],n+2],n+3]
= g[[0,d],n+3]
= (n+3)+(n+3)+1
= 2n+7

#=> Generally g[[[0,d],1,k],n] = 2n+4k+3

g[[[0,d],2,1],n]
= g[f[[[0,d],2,1],n],n+1]
= g[[[[0,d],2,0],1,[0,d]],n+1]
= g[f[[[[0,d],2,0],1,[0,d]],n+1],n+2]
= g[[[[[0,d],2,0],1,n+1],0,[[0,d],2,0]]],n+2]
= g[f[[[[[0,d],2,0],1,n+1],0,[[0,d],2,0]]],n+2],n+3]
= g[[[[0,d],2,0],1,n+1],n+3]
= ...
= g[[[0,d],2,0],3n+6]
= g[f[[[0,d],2,0],2n+6],3n+7]
= g[[0,d],3n+7]
= (3n+7)+(3n+7)+1
= 6n+15


It quickly ramps up from there. Some points of interest:

g[[[0,d],3,[0,d]],n] ≈ Ack(n,n), the Ackermann function
g[[[0,d],3,[[0,d],0,0]],63] ≈ Graham's number
g[[[0,d],5,[0,d]],n] ≈ G(2^^n), where 2^^n = n applications of 2^x, and G(x) is the length of the Goodstein sequence starting at x.


Eventually introducing more arguments of the f function as well as more cases for the array allows us to surpass most named computable notations. Some particularly known ones:

g[[,3,[0,d]],n] ≈ tree(n), the weak tree function
g[[[,3,[0,d]],2,[0,d]],n] ≈ TREE(n), the more well-known TREE function
g[[[[0,d]],5,[0,d]],n] >≈ SCG(n), sub-cubic graph numbers
g[[],n] ≈ S(n), Chris Bird's S function

• Ordinal explanation? – CalculatorFeline Dec 9 '17 at 2:55
• Is this your largest defined number yet? It appears so! – ThePlasmaRailgun Sep 28 '19 at 3:47

# Pyth, fψ(ΩΩ)+ω2+183(~25627!)

=QC.pGL&=^QQ?+Ibt]0?htb?eb[Xb2yeby@b1hb)hbXb2yeb@,tb&bQ<b1=Y_1VQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQVQ.v%%Fms["*s[.v"*\\^2d"\"%s"*\\^2d"\"")Qs["=Y.v+*"*\\^2Q"\"*3]"*\\^2Q"\"Q\YuyFYHpQ)


Requires any non-empty input, but the value thereof is not used.

Explanation (for the new and actually-reasonably-scoring version):

=QC.pG                   Sets the value of the autofill variable to app. 256^27!
27! ~= the number of characters in the string
containing all permutations of the alphabet.
We interpret that string as a base-256 number.
L                  Define a function y(b,global Q):
&=^QQ             Set Q to Q^Q and:
?+Ibt]0           If (?) the variable (b) is (I)nvariant on (+)adding itself
to the empty array (i.e. if it's an array itself):
?htb        If the second element of b is not 0:
?eb         If the last element is not 0
[Xb2yeby@b1hG)   return [b with its last element replaced with y(b[-1]), y(b), b]
hb                 else return b
Xb2yeb     else return b with its last element replaced with y(b[-1])
@,tb&bQ<b1      If b isn't an array,return:
either b-1 if it's a standard ordinal (1 or more)
or Q if b is ω
or 0 if b is 0
=Y_1                          Set the global variable Y to -1 (representing ω)
VQ                        Q times, do (the rest of the explanation):
VQVQ....VQ               Iterate from 0 to Q-1 183 times, each subiteration
reading the most recent value of Q when it starts:
.v%%Fms["*s[.v"*\\^2d"\"%s"*\\^2d"\"")Q
Iterate from 0 to Q-1 Q times, each subiteration
reading the most recent value of Q when it starts:
s["=Y.v+*"*\\^2Q"\"*3]"*\\^2Q"\"Q
Y = [Y,Y,Y] Q times, stacking with previous iterations.
uyFYHpQ)                    Run y_x(Y) for x incrementing until y_(x-1)(Y)=0


It's very hard for me to compute the size of this, mostly because it's late in the day and I'm not super familiar with fast-growing hierarchies or how I'd even go about trying to figure out how many times Q goes through the y() wringer. While I now know more about ordinals, I still have no idea how to calculate the value of the ordinal represented by the recursive definition in my program. I'd join the Discord server, but that's under a pseudonym I'd rather not be linked to my real name.

Unfortunately, because I know relatively little about said fast-growing hierarchies, I'm likely to have already lost to the Ruby answer. It's hard for me to tell. I may have beaten the Ruby answer, but I'm not 100% sure. ¯\_(ツ)_/¯

• If I understand correctly, your score is probably somewhere in the ballpark of 27^^^27^^27^^4, or f<sub>4</sub>(27^^27^^4)) ≈ f<sub>4</sub>(f<sub>3</sub>(f<sub>3</sub>(19))). – Simply Beautiful Art Nov 25 '17 at 15:22
• I made a small change that I should have thought of yesterday, but somehow didn't - making y recurse to operate on y(Q-1) instead of operating just on Q. How does this affect the score? – Steven H. Nov 25 '17 at 19:13
• I'm not entirely sure what's going on. Does y(Q) = L(y(Q-1)), per se? – Simply Beautiful Art Nov 25 '17 at 19:18
• I think we'll have better luck doing this in a chatroom. – Steven H. Nov 25 '17 at 19:22
• @SimplyBeautifulArt Its probably best not to use fast growing hierarchy notation for this, since its kind of small. – PyRulez Nov 27 '17 at 1:25

# Pyth, f3+σ-1+ω2(25626)

Where σm[n] is the Busy Beaver function Σ of order m called on n: σm[n] = Σm(n). The order -1 is to denote that the Busy Beaver here is not being called on a true Turing Machine, but rather an approximation with a finite wrapping tape of Q elements. This allows the halting problem to be solvable for these programs.

=QCGM.x-Hlhf!-/T4/T5.__<GH0M.x+Hlhf!-/T4/T5._>GHlGL=.<QCm.uX@[XN2%h@N2l@N1XN2%t@N2l@N1XN1X@N1@N2%h@@N1@N2l@N1XN1X@N1@N2%t@@N1@N2l@N1XW!@@N1@N2N2nFKtPNXW@@N1@N2N2gFK)@hNeN3%heNlhNd)bLym*F[]d^UQQUQUld)^U6QJ"s*].v*\mQ"
.v+PPPP*JQ"+*\mQ\'


The TL;DR is that this creates all possible BrainF**k programs of length Q, runs them in an environment where the maximum value of an integer is Q and the tape length is Q, and compiles all the states from these operations together to append (that's the 3+) to Q, repeating the above on a scale of fω2.

I still have ~half the characters to work with if I wanted to do something more, but until we figure out where this is I'll just leave it as is.

• I made a sorta better explanation of σ in the leaderboard. – Simply Beautiful Art Dec 13 '17 at 1:54
• It doesn't look to me like this particular Busy Beaver function an be that fast growing. With a limit of Q integers between 0 and Q, there are only (Q+1)^Q possible tapes, and Q possible positions in the program, so there can be at most Q*(Q+1)^Q possible states of a running program. So a program must halt within Q*(Q+1)^Q steps or not at all. The number of possible programs is also limited by an exponential upper bound. So it looks to me like this Busy Beaver function has an exponential upper bound, and the final function will be on the order of $f_{\omega^2}$. – Deedlit Dec 13 '17 at 6:17

# python, f3(f3(141)), 512 bytes

import math
def f(x):
return math.factorial(x)
x=9
for j in range(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))):
x=f(x)
print x


This isn't really a valid answer, but I wanted to post it anyway. A quick rundown:

import math # imports the factorial function
def f(x):
return math.factorial(x) # shortens the factorial operation
x=9 # sets x to highest available number
for j in range(f(...f(x)...)): # repeats * A LOT *
x=f(x) # does the factorial of x
print x # outputs the result


Anyway, I don't know if this answer technically legal, but it was fun to write. Feel free to edit any errors you find in the code.

• I think this is f_3(9), and it is definitely legal. You'd get a far larger number by nesting even for j in range(f(x)): for j in range(f(x)): x = f(x), though. Join us in chat to discuss why! – Steven H. Dec 12 '17 at 6:22
• Why isn't it a valid answer? – Simply Beautiful Art Dec 12 '17 at 9:11
• I didn't quite get the question, so I just made what I thought was right. – i.. Dec 13 '17 at 0:34

# Ruby, probably ~ fω+35(9999)

G=->n,k{n<1?k:(f=->b,c,d{x=[]
c<G[b,d]?b-=1:(x<<=b;c-=G[b,d])while c>=d
x<<[c]}
x=f[n-1,k,b=1]
(b+=1;*u,v=x;x=v==?u:v==[*v]?u<<[v-1]:u+f[n-1,G[v,b]-1,b])while x
b)};(n=9**9**99).times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n.times{n=G[n,n]}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}};p n


Try it online!

### Approximate Math Explanation:

The below is approximately equal to the program above, but simplified so that its easier to understand.

G(0,k) = k is our base function.

To evaluate G(n,k), we take k and write it as G(n-1,1) + ... + G(n-2,1) + ... + G(0,1).

Then change all of the G(x,1)'s into G(x,2)'s and subtract 1 from the entire result.

Rewrite it in the above form using G(x,2), where x<n, and leave the remainder at the end. Repeat, changing G(x,2) to G(x,3), etc.

When the result reaches -1, return the base (the b that would be in G(x,b).)

## Examples:

### G(1,1):

1: 1 = G(0,1)
2: G(0,2) - 1 = 1
3: 1 - 1 = 0
4: 0 - 1 = -1      <----- G(1,1) = 4


### G(1,2):

1: 2 = G(0,1) + G(0,1)
2: G(0,2) + G(0,2) - 1 = G(0,2) + 1
3: G(0,3) + 1 - 1 = G(0,3)
4: G(0,4) - 1 = 3
5: 3 - 1 = 2
6: 2 - 1 = 1
7: 1 - 1 = 0
8: 0 - 1 = -1      <----- G(1,2) = 8


### G(1,3):

1: 3 = G(0,1) + G(0,1) + G(0,1)
2: G(0,2) + G(0,2) + G(0,2) - 1 = G(0,2) + G(0,2) + 1
3: G(0,3) + G(0,3)
4: G(0,4) + 3
5: G(0,5) + 2
6: G(0,6) + 1
7: G(0,7)
8: 7
9: 6
10:5
11:4
12:3
13:2
14:1
15:0
16:-1      <----- G(1,3) = 16


### G(2,5):

1: 5 = G(1,1) + G(0,1)
2: G(1,2) + 1
3: G(1,3)
4: G(0,4) + G(0,4) + G(0,4) + G(0,4) + G(0,4) + G(0,4) + G(0,4) + 3
5: G(0,5) + G(0,5) + G(0,5) + G(0,5) + G(0,5) + G(0,5) + G(0,5) + 2
6: G(0,6) + G(0,6) + G(0,6) + G(0,6) + G(0,6) + G(0,6) + G(0,6) + 1
...
1024: -1      <----- G(2,5) = 1024


Doing some math, I found that

G(1,n-1) = 2ⁿ
G(2,n+6) ~ 2^G(2,n),  large enough n.


And beyond that it tends to get a bit hairy.

In general, we have

G(n,k+G(n-1,1)) ~ G(n-1,G(n,k)), large enough n.


# Python 3, fωω+ω*ω(99999)

from functools import*
h=lambda a,x,b:h(h(a,x,b-1),x-1,a)if x*b else a+b
def f(*x):
if(any(x[:2]):return reduce(lambda y,z:h(z,y,f(x,x-1,*x[2:])),x[::-1])if x*xelse(f(x-1,f(x-1,x,*x[2:]))if x>xelse(f(x-1,f(*([x-1]*2+x[2:])),*x[2:])))
for a,k in enumerate(x):if k:return f(*[f(*[k]*a,k-1,*x[a+1:])]*a,k-1,*x[a+1:])
return 0
x,s,g,e,r,z=9**9**9**99,"f(*[%s]*%s)",lambda a,b:a%((b,)*a.count("%")),"x*=eval(\"%s\");","x","x=g(e,g(reduce(g,[s]*x,s),r));"
print(exec(z*x)or eval(r))


I'll get an explanation up soon.

# Python 3, ~ fε0(999)

N=9**9**9
def f(a,n):
if a==[]:return a[1:]
if a==[]:return[a[1:]]*n+a[1:]
return [f(a,n)]+a[1:]
a=eval("["*N+"]"*N)
n=2
while a:a=f(a,n);n+=1
print(n)


Try it online!

• N=9**9e99 should be slightly larger – fejfo Dec 19 '17 at 21:02
• than whose answer? – Leaky Nun Dec 19 '17 at 21:03
• I mean that if you replace the first like with N=9**9e99 the output should be slightly larger because 9e99>9**9. Ofcourse It is still your answer. – fejfo Dec 19 '17 at 21:05
• @fejfo I mean it wouldn't change my ranking. – Leaky Nun Dec 19 '17 at 21:06
• Does that matter? – fejfo Dec 19 '17 at 21:06

# Python 3, 323 bytes, g9e9(9)

exec("""a=x:9**x
n=a,f:x:a and n(a-1,f)(f(x))or x
c=n:l:l[~n](l)
e=x:n(x,c(0))([x,l:[a(l),n(*l)],c(0),l:[a(l),l(l[:2])]+map(i:l((l,i)),l[2:])]+list(map(c,range(a(x),1,-1))))
f=l:[l(l),e(l(l))(l)]
g=x:e(x)((x,f))((x,a))(x)
print(n(9e9,g)(9))""".replace('','lambda '))


Try it online!

## Explanation

Python 3 is a truly recursive language, this means that not only can a function call itself, a function can also take other functions as input or output functions. Using functions to make themselves better is what my program is based on.

f=lambda x,a:[a(x),e(x)((x,a))]

### Definition

a(x)=9^x
b(x,f)=a(x), f^x
c(n)(*l)=l[~n](l)
c(0)=c0 <=> c0(…,f)=f(…,f)
d(x,b,c,*l)=a(x), c0(x,b), b(x,c0), b(x,f) for f in l
e(x)=c0^x(x,b,c0,d,c(a(x)),c(a(x)-1),c(a(x)-2),…,c(3),c(2),c(1))
f(x,a)=a(x),e(a(x))(x,a)(x)
g(x)=e(x)(x,f)(x,a)(x)
myNumber=g^9e9(9)


### Definition explained

a(x)=9^x a is the base function, I chose this function because x>0=>a(x)>x which avoids fixed points.

b(x,f)=a(x), f^x b is the general improving function, it takes in any function and outputs a better version of it. b can even be applied to itself: b(x,b)=b^x b(x,b^x)=b^(x*x)

but to fully use the power of b to improve b you need to take the output of b and use it as the new b, this is what c0 does:

c0(…,f)=f(…,f)
c0(x,b^x)=b^x(x,b^x)>b^(9↑↑x)

the more general c(n) function takes the n last argument (starting from 0) so c(1)(…,f,a)=f(…,f,a) and c(2)(…,f,a,b)=f(…,f,a,b). *l means l is an array and l[~n] takes the n last argument

d(x,b,c,*l)=a(x), c0(x,b), b(x,c0), b(x,f) for f in l d uses c0 to upgrade b and b to upgrade all of the other input functions (of which there can be any amount because of the list)
d(x,b,c,d)>9^x,b^x,c^x,d^x and d²(x,b,c,d)>a²(x), b^(9↑↑x), c^(9↑↑x), d^(9↑↑x)

but d gets even better if you combine it with c:
c0²(x,b,c0,d)=d^x(9^x,b^x,c0^x,d^x)=… c0(x,b,c0,d,c1)=c1(x,b,c0,d,c1)=d(x,b,c0,d,c1)=9^x,b^x,c0^x,d^x,c1^x c0²(x,b,c0,d,c1)=c0(9^x,b^x,c0^x,d^x,c1^x)=c1^x(9^x,b^x,c0^x,d^x,c1^x)=…

the more c(x) you add at the end the more powerful it becomes. The first c0 always remains d: c0(x,b,c0,d,c4,c3,c2,c1)=c1(…)=c2(…)=c3(…)=c4(…)=d(x,b,c0,d,cX,cX-1,…,c3,c2,c1)=…
But the second leaves iterated versions behind:

c0²(x+1,b,c0,d,c4,c3,c2,c1)
=c0(9^x+1,b^x+1,c0^x+1,d^x+1,c4^x+1,c3^x+1,c2^x+1,c1^x+1)
=c1^x(c2^x(c3^x(c4^x(d^x(9^x+1,b^x+1,c0^x+1,d^x+1,c4^x+1,c3^x+1,c2^x+1,c1^x+1)))))


When d^x is finally calculated c4 will take a much more iterated version of d the next time. When c4^x is finally calculated c3 will take a much more iterated version of c4,…
This creates a really powerful version of iteration because d:

1. Improves b using c0
2. Improves c0 using b
3. Improves all the layers of nesting using b The improves themselves improve, this means d becomes more powerful when it is iterated more.

Creating this long chain of c is what what e(x)=c0^x(x,b,c0,d,c(a(x)),c(a(x)-1),c(a(x)-2),…,c(3),c(2),c(1)) does.
It uses c0^x to bypass that c0 would just give d.
The  means that it will eventually return the second output of d^…. So b^….

At this point I couldn't think of any thing to do with e(x) to significantly increase it's output except increasing input.

So f(x,a)=a(x),e(a(x))(x,a)(x) uses the b^… generated by e(x) to output a better base function and uses that base function to call e(x) with a bigger input.

g(x)=e(x)(x,f)(x,a)(x) uses a final e(x) to nest f and produces a really powerful function.

### Fgh approximation

I will need help approximating this number with any sort of fgh.

Old version: fωω6(fωω5 (9e999)), Try it online! Revision history of explanation

• Actually, f_1(x) = x+x, but in the long run, this doesn't matter too much. – Simply Beautiful Art Dec 24 '17 at 0:41
• Could you explain your fundamental sequences a bit more? – Simply Beautiful Art Dec 24 '17 at 0:41
• @SimplyBeautifulArt ow yes I forgot to update that after I changed it from x*x. – fejfo Dec 24 '17 at 10:15
• @SimplyBeautifulArt My answer doesn't use any ordinals so it's hard for me to explain it with ordinals. All I really can do is give the definition of my functions and an approximation of the effect in the fgh. Example: a2(f_n)~=f_{n+1} – fejfo Dec 24 '17 at 10:16

# Ruby, fε02(5), 271 bytes

m=->n{x="";(0..n).map{|k|x+="->f#{k}{"};x+="->k{"+"y=#{n<1??k:"f1"};k.times{y=f0[y]};y";(2..n).map{|l|x+="[f#{l}]"};eval x+(n<1?"":"[k]")+"}"*(n+2)}
g=->z{t="m[#{z}]";(0...z).map{|j|t+="[m[#{z-j-1}]]"};eval t+"[->n{n+n}][#{z}]"}
p m[m][m][m][m][m][g]


Try it online!

This is based off of the m(n) map.

## Explanation:

m[f0][k] = f0[f0[...f0[k]...]] with k iterations of f0.

m[f0][f1][k] = f0[f0[...f0[f1]...]][k] with k iterations of f0.

m[f0][f1][f2][k] = f0[f0[...f0[f1]...]][f2][k] with k iterations of f0.

In general, m[n] takes in n+2 arguments, iterates the first argument, f0, k times onto the second argument, and then applies the resulting function onto the third argument (if it exists), then applies the resulting function onto the fourth argument (if it exists), etc.

### Examples

m[n↦n+1] = (((3+1)+1)+1 = 6

In general, m[n↦n+1] = n↦2n.

m[m[n↦n+1]] = m[n↦2n] = 2(2(2(3))) = 24

In general, m[m[n↦n+1]] = n↦n*2^n.

m[m]
= m[m[m[n↦n+1]]]
= m[m[n↦2n]]
= m[n↦n*2^n]
= (n↦n*2^n)[(n↦n*2^n)[n↦n*2^n(3)]]
= (n↦n*2^n)[(n↦n*2^n)]
= (n↦n*2^n)
= 402653184*2^402653184


In general, m[m][n↦n+1] = f_ω in the fast-growing hierarchy.

g[z] = m[z][m[z-1]][m[z-2]]...[m][m][n↦2n][z]


and the final output being

m[m][m][m][m][m][g]