Here's one from basic set theory. Let k be a cardinal and consider the operation "adding k", meaning
l |--> k+l
on cardinals. We know that this operation "stabilizes" to the identity after k, that is, for any l>k, we have l+k = l. Similarly, the "multiplying by k" operation,
l |--> l * k
stabilizes to the identity after k.
Everyone also knows that if l is an infinite cardinal then l^2 is equipotent to l, and more generally l^n is equipotent to n l for every natural number n. I.e. all the finite power functions stabilize to the identity at omega.
Well, obviously "exponentiation by omega" also stabilizes at some point, right? Like, l^omega is equal to l for sufficiently large l? Look, we probably already have the stabilization point at 2^omega.