In standard set theory (ZF) this kind of set is forbidden because of the [axiom of foundation](http://en.wikipedia.org/wiki/Axiom_of_foundation).

There are alternative axiomatisations of set theory, some of which do
not have an equivalent of the axiom of foundation. This is called
non-well-founded set theory. See e.g. [Aczel's anti-foundation_axiom](http://en.wikipedia.org/wiki/Aczel%27s_anti-foundation_axiom), where there is a unique set such that $x = \{x\}$.