Your goal is to implement the (asymptotically) fastest growing function within bounded code on a fictional CPU utilizing a quite limited, yet (probably) turing-complete instruction set.
The CPU utilizes unbounded RAM as well as two registers, the accumulator A and the program counter C, with words consisting of arbitrary integers, such that neither overflows nor underflows are possible. RAM is used to store data as well as code, allowing for self-modifying programs. Each instruction takes one parameter and therefore consists of two words; all instructions of your program are stored sequentially in RAM, starting at address 0. The following instructions can be used, P representing the parameter of the instruction:
|LOAD P||0||A := RAM[P]; C += 2|
|SAVE P||1||RAM[P] := A; C += 2|
|CNST P||2||A := P; C += 2|
|ADDT P||3||A += RAM[P]; C += 2|
|NEGA P||4||A := -RAM[P]; C += 2|
|JUMP P||5||C := P|
|JMPN P||6||If A <= 0 then C := P else C += 2.|
|HALT P||every other number||The program halts.|
At each step, the instruction at address C will be executed using the parameter stored at C + 1. Both A and C will be initialized to 0 at the start of a program's execution. The word at -1 is supposed to be your input which can be guaranteed to be non-negative, other words not storing any instructions initially contain 0. The number stored at -2 will be considered your program's output, which must also be positive in all but finitely many cases.
At the initial state, your program may not occupy more than the first 2048 words each storing integers between -2^64 and 2^64, however, during execution, there are no bounds. Of course, you don't have to write your program in bytecode, using some assembly equivalent or ultimately any other language is fine as well, as long as you provide some rules/translator and show the result does not exceed the given bounds.
Every answer should come with some rough argument showing the program always halts, as well as some approximate lower bound on its growth rate. As the given space might very well suffice for some extremely fast-growing functions, it might be helpful to utilize the slow-/fast-growing hierarchy, as it provides a relatively simple way to compare two answers. Answers will be ranked by lower bounds that can be shown to hold.