User blog comment:Pellucidar12/Proof length function/@comment-24920136-20161228041401/@comment-30754445-20161228053355

The standard "trick" is to use 2^^n instead of n.

But I don't think that using faster-growing functions would matter much. After all, we could always use our new fast-growing function f(n) and feed it into itself. Intuitively (although I can't prove it) this should lead to a case of diminishing returns very quickly.