I'm looking for good examples where double induction is necessary. What I mean by double induction is induction on $\omega^2$. These are intended as examples in an "Automatas and Formal Languages" course.

One standard example is the following: in order to cut an $n\times m$ chocolate bar into its constituents, we need $nm-1$ cuts. However, there is a much better proof without using induction.

Another example: the upper bound $\binom{a+b}{a}$ on Ramsey numbers. The problem with this example is that it can be recast as induction on $a+b$, while I want something which is inherently inducting on $\omega^2$.

Lukewarm example: Ackermann's function, which seems to be pulled out of the hat (unless we know about the primitive recursive hierarchy).

Better examples: the proof of other theorems in Ramsey theory (e.g. Van der Waerden or Hales-Jewett). While these can possibly be recast as induction on $\omega$, it's less obvious, and so intuitively we really think of these proofs as double induction.

Another example: cut elimination in the sequent calculus. In this case induction on $\omega^2$ might actually be necessary (although I'm not sure about that).

The problem with my positive examples is that they are all quite technical and complicated. So I'm looking for a simple, non-contrived example where induction on $\omega^2$ cannot be easily replaced with regular induction (or with an altogether simpler argument). Any suggestions?