User blog comment:ATAPORAE/Предел гугологии/@comment-119611-20150102154337/@comment-1605058-20150102155427

Busy Beaver is only the limit of what we call here "computable googology". There are many things beyond this, namely functions which aren't defined in terms of a strict procedure/algorithm which evaluates them. As a simple example, take your favorite programming language P, and define P(n) as largest number which can be output of program of P of length at most n. There is no real procedure which would give us value of this function (something like "simulate P on every input of length at most n" won't work, because some programs might not halt). But the point is, this largest number always exist, which makes the function well-defined. There are many ways to get even further - limit of googology, if exists, lies very far beyond Busy Beaver function.