Comment by cousin_it

BB_2(X) uses a different trick. It asks for the largest number computable by any program of length X that's allowed to call a BB oracle with arbitrary arguments. That's more powerful than any finite expression involving BB and computable functions, like BB(F(X)), BB(F(BB(X))), etc.

More generally, I expect there to not be any "meta-algorithm" for naming big numbers. Every big step will require a new insight. A similar problem is inventing notations for large ordinals: http://en.wikipedia.org/wiki/Ordinal_notation