Set containing set containing set containing... I was wondering if anyone knew whether or not in ZFC (or any other set theory) if the object,
$$\Bigg\{ \Big\{ \big\{ \{ \cdots \} \big\} \Big\} \Bigg\}$$
can exist.  That is, it is the set containing the set containing the set containing...ad infinitum.  Is this even a set?  Can it make sense to talk about such a thing?  I'm simply curious.
 A: The axiom of foundation (or regularity) explicitly bans your set. This axiom is independent of the other axioms of ZFC and, depending on your point of view is either just a technical convenience or a monster-barring axiom. Proponents of non-well-founded set theory like these monsters.
To see why your set gives rise to a violation of the axiom of foundation, note that (as explained on the wikipedia page), the axiom of foundation implies that that the membership relation is well-founded: i.e., there is no infinite sequence of sets $A_1, A_2, \ldots$ such that $A_1 \ni A_2 \ni A_3 \ldots$, but your set gives rise to such a sequence.
A: It depends very much on how you handle the informal "..." and the words ad-infinitum.  The short answer is that the notation is not clear enough to answer whether it is a set or a class.
If you define your construct to be "a thing which has the property of having 1 element, which is itself," you have constructed a class, not a set, because a set cannot have those properties in most set theories.
On the other hand, if you construct it using a sequence S which starts with $S_0=\emptyset$ and is recursively constructed according to $S_{n+1}=\{S_n\}$, and define the thing you want to be $S_\omega$, then what you are constructing is not all that far from the standard set theory construction of natural numbers ($S_{n+1}=S_n \cup \{S_n\}$)
