I asked the same question a while ago, https://math.stackexchange.com/questions/613946/xx-y-given-y...
Interesting analysis of the problem :-) it has been bugging me since high school!
Might be worth linking to the Lambert W function https://en.wikipedia.org/wiki/Lambert_W_function
This was the best comment I'd reckon:
> As you probably saw, the solution of x^x = y is x = Log(y) / W[Log(y)]. Now you take us in a highly philosophical debate. What is the limit of elementary functions ? Thanks for this question. Cheers. – Claude Leibovici