What are the nontrivial theorems one can prove about natural numbers which need transfinite induction? By "need transfinite induction" I mean one can show that the statement is not provable in arithmetic which only allows induction up to $\omega$. In particular, are there any statements which only state a claim about natural numbers but which require the use of infinite objects in their proof? The statement should not be metamathematical in nature (proof of consistency etc) or involve any ordinals or cardinals (i.e. infinite objects). Goodstein's theorem was mentioned as one such possibility in this MO question
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 $\omega_1$.