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.
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.
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 ($S_{n+1}=S_n \cup \{S_n\}$)