[perl4ever]

Like I said, I can't really follow this stuff, but maybe the ultimate would be the Busy Beaver function? Is there such a thing as a proof of how many functions can be between something computable and Busy Beaver, given some sort of constraints?

[Gondolin]

Yes, the Busy Beaver function grows faster than the functions I gave, basically it behaves like f_{w^1_CK} (although this does not really make sense because w^1_CK, the Church Kleen ordinal, is not recursive).

By "ultimate generalisation" I was speaking about a family of computable functions that ultimately grow faster than other functions you could cook up by hand: you put all the recursivity in the ordinals, they are there for that, and logicians are very good at constructing very large (recursive) ordinals.

If you just want a very very fast growing function (but that is still computable, unlike busy beaver), you can do the a light busy beaver version: Beaver(N)=the maximal output of all turing machines T with N state where there is a proof in ZFC of length <=N that T terminates.

This function grows much faster than the other functions I described (basically it is a f_{ordinal consistency of ZFC}. If you are interested in this kind of ideas, there is this reddit post that develops these ideas: https://www.reddit.com/r/math/comments/283298/how_to_compute...