What is this operator called? If $x \cdot 2 = x + x$
and $x \cdot 3 = x + x + x$
and $x^2 = x \cdot x$
and $x^3 = x \cdot x \cdot x$
Is there an operator $\oplus$ such that: 
$x \oplus 2 = x^x$
and $x \oplus 3 = {x^{x^x}}$?
Also, is there a name for such a set of operators ops where...
Ops(1) is addition
Ops(2) is multiplication 
Ops(3) is exponentiation
Ops(4) is $\oplus$
...and so on
Also, is there a branch of math who actually deals with such questions? Have these questions already been answered like 2000 years ago?

This operation  ${\rm Ops}(4)$ is called tetration, from the greek root tetra meaning four; it's also sometimes called a "power tower". There are also many further generalizations of this type of sequence; Knuth's up-arrow notation gives $a^{a^{a^a}}=a\uparrow\uparrow4$, so that $a\uparrow\uparrow n$ is the tetration operation. By adding more arrows you get pentation and so on, and the Conway chained arrow notation generalizes this still further.
FYI, for "to the power of-ation" the word you're looking for is exponentiation.
