Given $f^k(x) = f \circ f \circ f\circ ...(x)$ composed $k$ times,
Do the functional equations $f^k(x) = g(x)$, where $g(x)$ is a basic transcendental elementary function, for example, the inverse hyperbolic trigonometric functions, have elementary solutions?
e.g. $f \circ f (x) = \sinh^{-1}(x) $
This question was raised by a friend of mine with a well developed understanding of (Australian?) high-school mathematics, and some fundamentals of applied mathematics (e.g analysis), but not much else, so try to be as clear as this situation allows it.