In the usual presentation of Goodstein's theorem, the base is bumped up by the "add 1" function. Does the theorem still hold when we replace this function by a fast-growing one (e.g. Ackermann or busy beaver)? How far can we push this? For example, let's define $g\_0(n)$ to be the number of Goodstein iterations needed to reach 0 when we start with base 2 and seed $n$ (so that $g\_0(0)$ = 0). Then we can build a hierarchy of functions by defining $g\_{k+1}(n)$ as the number of Goodstein iterations needed to reach 0 with seed $n$ and base-bumping function $g\_k$ ($k$ = 0, 1, ...), continuing through the ordinals by diagonalization at each limit ordinal. Surely it's got to break down when we go past $\varepsilon\_0$, if not long before that! 



How fast can the base-bumping function in Goodstein's theorem grow?
As long as your fast-growing "base-bumping" function still takes every natural number to a natural number (instead of, say, an infinite ordinal)--and the busy beavers do--the Goodstein iterations are still upper-bounded by the strictly-decreasing sequence of ordinals in "base" $\omega$, which must be of finite length as a decreasing sequence in a well-ordered set.


