Haskell, 100 bytes, score ≈ 10↑↑65503

f!x|x<' '=f|q<-(!pred x),r<-(q$q f)[x]=foldl(.)q[q|_<-r,_<-r]f
main=print$(\x->x++x)!'�'$[pred ':']

The special character (2^16 - 3 ascii) counts as 2 bytes. pred ':' is equal to '9'.

I'll be severely impressed if someone can tell me what my score is.

Arguably more readable version in C-like psuedocode:

myFunc = (n => f => {
  if (n == 0) return f

  q = myFunc (n-1)
  r = q(q(f)) ("a")

  fa = q
  for (i=0; i < r.length^2; i++)
    fa = (func => q (fa (func)))

  return fa (f)
})

print myFunc(65502)(str => str + str)("9")

Haskell, 100 bytes, score ≈ 10↑↑65503

f!x|x<' '=f|q<-(!pred x),r<-(q$q f)[x]=foldl(.)f[f|_<-r,_<-r]
main=print$(\x->x++x)!'�'$[pred ':']

The special character (2^16 - 3 ascii) counts as 2 bytes. pred ':' is equal to '9'.

Haskell, 100 bytes, unknown score

f!x|x<' '=f|q<-(!pred x),r<-(q$q f)[x]=foldl(.)q[q|_<-r,_<-r]f
main=print$(\x->x++x)!'�'$[pred ':']

The special character (2^16 - 3 ascii) counts as 2 bytes. pred ':' is equal to '9'.

I'll be severely impressed if someone can tell me what my score is.

Arguably more readable version in C-like psuedocode:

myFunc = (n => f => {
  if (n == 0) return f

  q = myFunc (n-1)
  r = q(q(f)) ("a")

  fa = q
  for (i=0; i < r.length^2; i++)
    fa = (func => q (fa (func)))

  return fa (f)
})

print myFunc(65502)(str => str + str)("9")

Haskell, 100 bytes, score ≈ 10↑↑65503

f!x|x<' '=f|q<-(!pred x),r<-(q$q f)[x]=foldl(.)q[q|_<-r,_<-r]f
main=print$(\x->x++x)!'�'$[pred ':']

The special character (2^16 - 3 ascii) counts as 2 bytes. pred ':' is equal to '9'.

I'll be severely impressed if someone can tell me what my score is.

Arguably more readable version in C-like psuedocode:

myFunc = (n => f => {
  if (n == 0) return f

  q = myFunc (n-1)
  r = q(q(f)) ("a")

  fa = q
  for (i=0; i < r.length^2; i++)
    fa = (func => q (fa (func)))

  return fa (f)
})

print myFunc(65502)(str => str + str)("9")

Haskell, 100 bytes, unknown score

x!f|x<' '=f|q<-(pred x!),r<-q$q$q f[x]=foldl(.)q[q|_<-r,_<-r]f
main=print$'�'!(\x->x++x)[pred ':']

The special character (2^16 - 3 ascii) counts as 2 bytes. pred ':' is equal to '9'.

I'll be severely impressed if someone can tell me what my score is.

Haskell, 100 bytes, unknown score

f!x|x<' '=f|q<-(!pred x),r<-(q$q f)[x]=foldl(.)q[q|_<-r,_<-r]f
main=print$(\x->x++x)!'�'$[pred ':']

The special character (2^16 - 3 ascii) counts as 2 bytes. pred ':' is equal to '9'.

I'll be severely impressed if someone can tell me what my score is. 

Arguably more readable version in C-like psuedocode:

myFunc = (n => f => {
  if (n == 0) return f

  q = myFunc (n-1)
  r = q(q(f)) ("a")

  fa = q
  for (i=0; i < r.length^2; i++)
    fa = (func => q (fa (func)))

  return fa (f)
})

print myFunc(65502)(str => str + str)("9")