How to denote "powers" of a function? I'm working with functions themselves, and I have learned that functional powers mean composition so:

$f^3 = f \circ f \circ f$

But I'm looking for something that means $fff$. So $(fff)(x) = (f(x))^3$
Is there a de facto standard notation I can use when mixing the two? Preferably without having to write out the $(x)$ part?

It seems we have a good deal ambiguity here. Right at the question title: The power of a function may mean two things. Sometimes even the circle is omitted to make it look like multiplication.
I've already seen the following, which I adopted: $$\begin{align} f^n(x) &amp;= (f(x))^n \\ f^{(n)}(x) &amp;= \frac{{\rm d}^nf}{{\rm d}x^n} \\ f^{\circ n}(x) &amp;= (f \circ f \circ \cdots \circ f)(x)\end{align}$$
However $f^{\circ n}$ does not seem to be standard, so you should always warn the reader when using it.




  The power of a function may mean two things. Sometimes even the circle is omitted to make it look like multiplication.


That's because the collection of all bijections from a set to itself form a group with composition of functions - and the operation in a group is usually seen as multiplication.
The way i read it, raising a number gives a power of the measure, so

$\sin^2 x$ corresponds to x sin square, and one is supposed to read down the tower of powers, not up them.

Multiple application, like $\sin \sin x$ = x, sin, sin, is not that common, and is best served by sequencing the terms from inner right to outer left.