TREE(3) is indistinguishable from zero compared to Loader's number[1], which basically takes every bit pattern up to some n and expresses this as a program in the Calculus of Constructions. This system is a bit weaker than being Turing complete, but the programs do always terminate (this makes the number computable compared with, say, the Busy Beaver number which does a similar thing with Turing complete programs).
It also has the geek cred of being represented by an obfuscated C program (the unobfuscated verson is also available[2]).
It also has the geek cred of being represented by an obfuscated C program (the unobfuscated verson is also available[2]).
[1] http://googology.wikia.com/wiki/Loader%27s_number
[2] https://github.com/rcls/busy