TITLE: Expressions for the inverse function of f(x) = ln(x)e^x
QUESTION [9 upvotes]: Can the inverse of $ ln(x)e^x $ be finitely expressed in terms of the Lambert-W function or any other well-known transcendental functions? It is clear that a closed-form elementary function expression is unreachable. 
The reason I ask is in pondering on the links between the inverse Lambert-W and some naturally arising functions of similar forms. Recall that the Lambert-W, a transcendental function, is defined as $ W(xe^x) = x. $
It is natural then to consider the inverse of functions such as $ g(x) = xe^{e^x} $ and those with further exponentiation. With a simple transformation $ z= e^x $ we can reduce $ g(x) $ to the form  $ ln(z)e^z $  as originally posed. So the broader question arises: are there tangible algebraic links between the inverses of the set
$$ {xe^x, xe^{e^x},xe^{e^{e^x}}}... $$

REPLY [8 votes]: These are so-called hyper-Lambert functions, see 
On some applications of the generalized hyper-Lambert functions.