Here is an example from group theory.
The automorphism tower of a group G is obtained by iteratively computing the automorphism group:
$$G\to \text{Aut}(G)\to \text{Aut}(\text{Aut}(G))\to\cdots$$
Each groups maps homomorphically into the next by mapping an element $g$ to conjugation by that element. One may therefore continue the iteration transfinitely by taking a direct limit to get the group $G_\omega$ at $\omega$, and continue the process. At successor stages, take the automorphism group; at limit stages, take the direct limit of the resulting system. The question is whether the process ever terminates, whether one ever arrives at a group that is isomorphic to its automorphism group by that natural map. Such a group is complete, having trivial center and no outer automorphisms.
Wielandt (1939) proved that the automorphism tower of every finite centerless group terminates in finitely many steps. Hulse (1970) proved that the automorphism tower of any centerless polycyclic group terminates in a countable ordinal number of steps.
Simon Thomas (here at MO) proved (1985) in general that the automorphism tower of any centerless group $G$ terminates before stage $(2^{|G|})^+$ many steps.
This bound on the height of the automorphism tower is strictly larger than the continuum, even when the size of the group is not, and so it seems to be an example of the desired phenomenon. (There is a set-theoretic sense (Just, Shelah and Thomas) in which one cannot expect to prove a better bound.)
Thomas' papers are available on his web page.
Meanwhile, in the case of non-centerless groups, I proved that every group has a terminating transfinite automorphism tower (see Proceedings AMS 126 (1998)). The proof proceeds by showing that every automorphism tower leads eventually to a centerless group, and then appeals to Thomas' theorem. There is also an easy survey article available.
For general groups, the best known upper bound for the height of the automorphism tower is essentially the next inaccessible cardinal. Even for finite groups, no reasonable upper bound is known in the general (non-centerless) case.
The topic was also discussed at this MO question.