Inverse of $y=xe^x$
Is there a closed form solution to the equation: $a e^a$ = 1 ?
In general, the solution to the equation $x=a e^a$ for $a$ is given by the Lambert W function, so in this case the solution is just $W(1)$, the so-called Omega Constant. $W(x)$ is not an elementary function, and can not be written in terms of more elementary functions. Whether or not the $W$ function counts as closed-form is up to you, as closed form means different things to different people.
As the Mathworld page above doesn't include any further information, it probably is not known if $W(1)$ itself has a closed form expression in terms of simpler functions. However, since negative results of this type are typically difficult to prove and there's no reason to expect such an expression, I'd strongly bet not even though I'd be surprised at such a proof. For reference, $W(1) \approx 0.56714$.