I have been looking at Goodstein sequences.
A descriptive link is at
http://mathworld.wolfram.com/GoodsteinSequence.html

Does anyone know when you reach 0 if you start from 4?  My estimate is
when the base is 3*2^(3*2^27+27)-1 which is a little less than
7*10^121210694 or rather more than 10^(10^8).

Essentially to calculate a(n+1), write a(n) in the hereditary
representation base n, then bump the base to n+1, then subtract 1.

You start with a number (for example 4) and put it in the heridiatary
representation base 2 (i.e. 2^2), bump it to base 3 by replacing the
2s by 3s (i.e. 3^3) and subtract 1 (to get 26).  

The next step is to do the same (i.e. 2*3^2+2*3^1+2) bump it to base 4
(i.e. 2*4^2+2*4^1+2) and subtract 1 (to get 41). etc.

For most starting points this seems grow explosively.  But starting
from 1 you get to 0 at base 3, starting from 2 you get to zero at base
5, and starting from 3 you get to 0 at base 7.  Goodstein’s theorem
(using transfinite induction) shows that you always get to zero. But
it takes a very long time. Does anyone know how long starting from 4?