The best base is 10... Let's reach it!
Mathematica, 56 bytes

#//.x_/;(b=Max[d=IntegerDigits@x]+1)&lt;11:&gt;d~FromDigits~b&amp;


Try it online! (Using Mathics.)

I thought I'd check out what the sequence looks like:



And here is a plot of the number of steps it takes to find the result:



(Click for larger versions. See the revision history for plots only up to n = 1000.)

Looks like a very interesting mixture of large scale structure and fine scale chaos. I wonder what's up with the wider gaps around 30,000 and 60,000.