Is there a notation for repeatedly applying a function? There is notation for applying both multiplication and addition a set amount of times (can be infinite): $\sum$, $\prod$ but there is no notation (to my knowledge) of repeating a function. Whenever I need this I use this notation: $R_{n}^{k} f(a) $ where $k$ is the amount of times it is repeated, $a$ is where the function is repeated, and $n$ is the input. For example: $$R_{n=\sqrt{2}}^{\infty} n^{a}=2$$
If anyone knows of the proper notation for this it would be greatly appreciated.

A common notation is $f^k(a)$, so
$$f^0(a)=a, f^1(a)=f(a), f^2(a)=f(f(a)), f^3(a)=f(f(f(a))),\dots.$$
You can see this notation used for $k=-1$ to denote the inverse of a function, e.g. $\sin^{-1}(x):=\arcsin(x)$.
For some functions, this notation is not very good; for example, $\sin^2(x)$ is usually defined as $\sin(x)^2$ instead of $\sin(\sin(x))$, since the former is a more useful function. It's a good idea to define this notation anywhere you might want to use it, and if there could be confusion, use something related like $f^{(k)}(a)$ (which can also be confused with the $k$th derivative of $f$) or $f^{\circ k}(a)$ (with the latter denoting "$f$ composed with itself $k$ times").
