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?

It is known as a tetration, and it is normally written as $^na$ where n is the height of the power tower. It is the forth hyperoperation.
The zeroth hyperoperation is the successor function, and the first is the zeroth hyperoperation iterated, and so on
A more general way to define the nth hyperoperation is, using the notation, $H_n(a,b)$ where n is the nth hyperoperation, 
${\displaystyle H_{n}(a,b)={\begin{cases}b+1&{\text{if }}n=0\\a&{\text{if }}n=1{\text{ and }}b=0\\0&{\text{if }}n=2{\text{ and }}b=0\\1&{\text{if }}n\geq 3{\text{ and }}b=0\\H_{n-1}(a,H_{n}(a,b-1))&{\text{if }n\in\mathbb{N},n>3}\end{cases}}}$
Some notations for hyperoperations are(for $H_n(a,b)$:


*

*Square bracket notation: $a[n]b$

*Box notation: $a{\,{\begin{array}{|c|}\hline {\!n\!}\\\hline \end{array}}\,}b$

*Nambiar's notation  : $a\otimes ^{n-1}b$

*Knuth's up arrow notation: $a\uparrow^{n-2}b$

*Goodstien's notation: $G(a,b,n)$

*Conway's chained arrow notation: $a\rightarrow b\rightarrow (n-2)$

*Bowers exploding array function: $\{a,b,n,1\}$

*Original Ackermann function: ${\begin{matrix}\phi (a,b,n-1)\ {\text{ for }}1\leq n\leq 3\\\phi (a,b-1,n-1)\ {\text{ for }}n\geq 4\end{matrix}}$

