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: $$\begin{align} f^n(x) &= (f(x))^n \\ f^{(n)}(x) &= \frac{{\rm d}^nf}{{\rm d}x^n} \\ f^{\circ n}(x) &= (f \circ f \circ \cdots \circ f)(x)\end{align}$$ 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 most common notation I've seen for $n$-fold composition is $$f(f(\ldots f(x)\ldots ))=f^{n}(x)$$ However this is generally always accompanied by a remark explaining that this is what the notation means. I would recommend you include such a remark.

I'm pretty sure there isn't a standard notation for raking a function to the power $n $, but again if you define some notation in the text then you're unlikely to be criticised.

FWIW, $f\,f\,f$ is also not quite unabiguous: it could be read as $f(f(f))$, or $f\:(f,f)$ – though both are really a bit strange and only make sense for polymorphic / higher-order functions.

Do use $f^n$ if you need this a lot, but also make some quick clarification about it. Or use something completely different – how about $\prod^n f$? That should be pretty unambiguous (at least unless you also have a symbol $\Pi$ around...).