@El'endiaStarman Probably only if I had to use it over and over again, and I'd explicitly stat it. Usually when you just have an exponent like in cos^2(x) you mean (cos(x))^2. Repeated composition is not used very often, but if there is the context where it is used a lot, you can of course do that, but I'd explicitly say so. Another clearer notation I like for repeated composition is f^{\circ n}(x) for f(f(...f(x)...))
@ConorO'Brien That is used a lot in my experience. Especially when the argument of the function is simple and you omit the parenthesis: e.g. cos^2 x means (cos(x))^2, just to avoid confusion with cos(x^2).