The Hydra Game is another nice example where ordinals are needed. More generally, ordinals and transfinite induction appear a lot in game theory, and in related subjects like automata theory.

For instance when studying thin trees (infinite trees with a countable number of branches), each thin tree can be mapped to a (countable) ordinal that describes its complexity in some sense, and proving results on thin trees can involve transfinite induction up to $\omega_1$.