  Goodstein's principle is arguably the first purely number-theoretic statement
known to be independent of Peano arithmetic. It involves sequences of natural
numbers which at first appear to grow very quickly, but eventually decrease to
zero. These sequences are defined relative to a notation system based on
exponentiation for the natural numbers. In this article, we explore notions of
optimality for such notation systems and apply them to the classical Goodstein
process, to a weaker variant based on multiplication rather than
exponentiation, and to a stronger variant based on the Ackermann function. In
particular, we introduce the notion of base-change maximality, and show how it
leads to far-reaching extensions of Goodstein's result.
