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I'm not sure if said set exist or whether it is unique, but what name could I use to find more about it and what kind of interesting properties does it have?

Clarification edit: I meant a set $V$ such that $V = V^V$. I am curious about the concept and would like to read more about it; I don't mind which formalism this is in. (From the answers, I understand that this isn't possible in most.)

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    If you like applying functions to themselves, you should check out something like this paper on finite left-distributive algebras and embedding algebras: http://arxiv.org/abs/math/9209202v12012-09-04

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Edit: Apparently you want to know the name for the set $V$ which satisfies $V=V^V$. I see no reason to believe a priori that such a set exists or is unique. However, we can argue by cardinality that $|V|=1$ if $V$ exists. Then it becomes an issue of whether you mean $V=V^V$ or $V\cong V^V$. The first is impossible, since the function $f:v\to v$ which maps the single element $x$ to itself is not literally the same as $x$. The second applies to all singletons.

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    @AntonGolov The$r$e's also $t$he issue of whe$t$her you mean equal or isomorphic.2012-09-04
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If by $V$ you mean "the von Neumann universe," that is, the class of all sets, then $V$ is not a set, let alone $V^V$.

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    @HenningMakholm: Thanks, fixed.2012-09-04
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Probably not what you intended, but since you explicitly don't mind which formalism:

In set theory with Aczel's anti-foundation axiom, there is exactly one set $x$ such that $x=\{\langle x,x\rangle\}$, and its singleton $\{x\}$ then satisfies your condition.

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Such a set does not exist.

We must have $|X| = 0$ or $X = |1|$, or else $X^X$ has more elements than $X$. If $X = 1$, say $X = \{x\}$, then $x = (x,x)$, which is impossible. If $X = 0$ this is also impossible, because $\emptyset \in \emptyset^\emptyset$ as Arthur pointed out.

Here is a simpler proof not involving Cantor's theorem. If $X = X^X$ and $X$ is nonempty them by the axiom of regularity $X$ has an element $x$ with minimal rank. However, $x$ is a (nonempty) function $X \to X$, so it contains some ordered pair $(y,z)$ of elements of $X$, and then $y$ and $z$ have smaller rank than $x$, a contradiction.