(This question is partly inspired by What is inter-universal geometry?.)
I have absolutely no background in Teichmuller theory or any related subject, but what I can follow of Mochizuki's description of inter-universal Teichmuller theory fascinates me. In particular, I'm very interested in what I perceive as his general claim* that an ill-founded set theory would represent certain mathematical objects more intuitively, and I'd like to get a handle on this, independently of his specific work.
I'm looking for reasonably natural mathematical structures which, in some sense, "contain themselves" as an element (or element of some element, or etc.). (This is obviously a fairly soft question, and I apologize in advance if it is inappropriate for MO.)
To clarify what I mean, I know of basically two specific examples. The first is the universal set in the set theory $NF$; the second is the set of (isometry types of) compact metric spaces of diameter $\le 1$, which under the Gromov-Hausdorff metric forms a compact metric space of diameter $\le 1$. Both of these, I imagine, can be expanded: the former could be replaced with any other consistent notion of universal set, or universal category; and my understanding is that there are a number of moduli spaces which are also naturally elements of themselves. This is the sort of thing I'm looking for: well-defined mathematical structures which can naturally be thought of as containing themselves as "elements," or "points," etc.
EDIT: what I say here about the Gromov-Hausdorff metric appears to be very wrong: see Nicola Gigli's answer below. Can this would-be example be fixed?
I'm especially interested in whether there are natural examples of the form $a_0"\in" . . . "\in" a_n"\in" a_0$ for $n>0$, since I know of no natural such example.
ADDED: An interesting observation is that - uniquely out of all examples and near-examples that I know - the Gromov-Hausdorff example above is not naturally "maximal" among its own elements. That is, there is no sense (that I'm aware of, at least) in which the space of all compact metric spaces of diameter $\le 1$ is the largest such metric space. This is obviously not the case for the universal set (in set theories which allow such objects), or the set of computable partial functions, or any variants of these. So a sub-question: does anyone have an example of a self-containing structure which is not somehow "maximal" amongst its elements?
*On, e.g., page 55 of Inter-universal Teichmuller Theory IV (http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf).