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](http://mathoverflow.net/users/4706/simon-thomas))
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](http://www.math.rutgers.edu/~sthomas/papers.html).

Meanwhile, in the case of non-centerless groups, I proved
that every group has a terminating transfinite automorphism
tower (see [Proceedings AMS 126
(1998)](http://www.ams.org/journals/proc/1998-126-11/S0002-9939-98-04797-2/home.html)).
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](http://front.math.ucdavis.edu/9808.5094)
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](http://mathoverflow.net/questions/5635/does-autaut-autg-stabilize).