The product logarithm (Lambert W) is the inverse of $\displaystyle f(z) = z e^z$. Wikipedia claims it is not elementary, but gives no reference. How do we know that is not elementary?
You can read a proof here: Algebraic properties of the Lambert W function from a result of Rosenlicht and of Liouville - Integral Transforms and Special Functions
It is not trivial, or even easy.
Tonio