$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$.
$(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$.
Is there an operator, such that if $x^{x^{\cdot^{\cdot^{x}}}}$, i.e. $x$ raised to the $x$th power $n$ times (with right associativity), we can write something like $x$ ¤ $n$, where the generic currency sign ¤ is a placeholder for the correct operator, if it exists?
Does such an operator exist? If so, what is the correct symbol? If not, why?
Is there a name for this operation? If so, what is it?
Depending on who you ask, you're going to get different responses.
You are referring to what is known as tetration. You can thank Reuben Louis Goodstein for this word, which is roughly "four iteration". It has many notations:
$$ ^{n}a, a {\uparrow\uparrow} n,a \rightarrow n \rightarrow 2, \text{uxp}_{a}n, a^{\underline{n}}, a^{(4)}n, \text{hyper}_4(a,n), . . . $$
For more, see Wikipedia: Tetration Notation.