Notation for repeated application of function

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.