Branch of mathematics that deals with repeated operations One interesting trait of subtraction is that it can introduce us to negative numbers using just positive whole numbers. For instance, $1-3=-2$. Division, similarly, can introduce us to a new set of numbers: decimals and fractions. For instance, $3/2=1.5$. Square roots can also show us a new set of numbers: imaginary. $\sqrt{-1}=i$.
The significance of this is that these operations are all inverses of operations that result from repeating a lower level function. If you could assign a number to this, you could say addition is 1. Multiplication is repeated addition, so it is 2 (it is defined as repeating a level 1 function). Exponentiation would be 3 (it is defined as repeating a level 2 function). Is there any branch of mathematics that deals with how properties of functions change as you go to higher levels? For example, exponentiation is not commutative ($3^2\neq2^3$). This is different from both addition and multiplication, both components of exponentiation (it is repeated multiplication, which is repeated addition). I believe I heard somewhere that as you get a level higher, one property changes, but that may have just been a theory of mine.
Is there any branch of mathematics that specifically deals with the properties of different levels of functions?
Another, perhaps deeper question involves stretching the limits of mathematics. If we do indeed have a branch of mathematics where people study functions in this way, is there such thing as a level 0 function? Some function that is more fundamental than addition? If there is a study of these operations, mathematics would find a way to answer all of these questions. what about a level -1? i? $\pi$?
 A: Without being a "branch of mathematics" at all, you find repeated operations like :
$A=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\cdots}}}}\tag{1}$
(belonging to the category of "continued fractions") or :
$$B=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}\tag{2}$$
the first question being whether they really define a number. In fact, if such is the case

*

*for (1), taking the inverse of both sides and adding one to them, we get the necessary condition $A=1+\dfrac{1}{1+A}$ which is equivalent to a quadratic equation with (one of its) solution(s) $A = \sqrt{2}$.


*for (2), a similar process gives $A=\sqrt{1+A}$, yielding as well a quadratic equation with (one of its) solution(s) $B = \frac12(1+\sqrt{5})\approx 1.618\cdots$, the golden ratio.
In fact both of them fall into a general category which is the rigorous definition of numbers as limits of recursive sequences :
$$u_{n+1}=f(u_n)$$
with a certain initial value $u_0$ with different functions $f$.
Not all $f$ and not all $u_0$ (far from that) giving rise to a convergent behavior...
