Notation for repeated application of function

If I have the function $f(x)$ and I want to apply it $n$ times, what is the notation to use?

For example, would $f(f(x))$ be $f_2(x)$, $f^2(x)$, or anything less cumbersome than $f(f(x))$? This is important especially since I am trying to couple this with a limit toward infinity.

In the course I took on bifurcation theory we used the notation $$f^{\circ n}(x).$$

You can use the notation $f^n$ to denote the composition of the function with itself $n$ times, though this may also mean the product of $f$ with itself $n$ times. Just make sure you define your notation at the start.

You could define the notation recursively as a sequence of functions.

Let $f_{n+1}(x) = f(f_n(x))$ for $n \geq 2$ with $f_1(x) = f(x)$.

Sequence notation of this type is so generic that the reader will be forced to consult your definition, which will avoid any possible misinterpretation.

If you take function iteration as a fold of self-composition, you can use a sum-like notation:

$\bigcirc^nf = \underbrace{f \circ \dots \circ f}_{n\:\text{times}}$

Where:

$\left({\bigcirc^0f}\right)(x) = x$

Granted, this is not very compact, and I would prefer to typeset the limit directly above the circle. That aside, it does combine tolerably with limit notation:

$\lim_{n\rightarrow\infty}\bigcirc^nf$

You can use $f^n(x)$ BUT be sure to tell the reader that you mean functional iteration, not $(f(x))^n$.