Notation of iterated composition of functions Let $f$ be a function from $A$ to $B$ such that the image $f(A)\subset A$. Is there a widely accepted notation for the expression
$$f\circ\left(f\circ\cdots(f\circ f)\right),$$
where $f$ composite with itself $n$ times? I failed to find a natural way to include the information $n$ in the notation.
 A: You wrote $f(B)\subset A$ but probably meant to write something like $B\subseteq A$ (or $f(A)\subseteq A$). 
I would plead for presenting $f$ not as a function $A\to B$ but as a function $A\to A$. 
This because $f\circ f$ is only properly defined if the codomain of $f$ coincides with the domain of $f$. 
The collection $M$ of functions $f:A\to A$ can be looked at as a monoid with the map prescribed by $a\mapsto a$ as identity and composition of functions as (associative) multiplication.
In a monoid $\langle M,\circ\rangle$ it is quite common to write $f^n$ for expression $f\circ\cdots\circ f$ containing $f$ exactly $n$ times.
So this is a nice justification for that notation.
A: A cleaner way to define the function above and the notation is presented below. 
Let $f: A \longrightarrow A$ and for $n \in \mathbb{N}$, let 
$$
f^n(x) :=
\begin{cases}
f(x),             & n=1 \\
f(f^{n-1}(x)),    & n>1.
\end{cases}
$$
A straightforward proof by induction shows that $f(f^n(x)) = f^n(f(x))$ for every natural number $n$.
Some have suggested the notation $f^{(n)}$ but, as noted in other comments, this is usually reserved for the $n^\text{th}$-derivative of $f$, which is also defined recursively.  
