Variable exponents are not algebraic.  There would be a
Taylor series solution for it.  But there is no algebraic
method.  However, just as was done in the case of all the
standard transcendental functions such as 

f(x) = sin(x), ln(x), exp(x), sinh(x),, etc.

a table of those were made up and printed in a book.  Then 
mathematicians were able to define things algebraically in 
terms of those non-algebraic (trancendental) functions. So 
a standard non-algebraic function could be derived, say, 
"floop(x)", and this could become as widely an accepted 
standard function as, say, sin(x), which could be defined 
as the sum of an infinite series. Then the solution to
x^x=2 could be defined in terms of floop(x).

But, let's face it, every standard trancendental function,
such as sin(x), ln(x), etc. can only be approximated by 
iterations.  

The solution to your equation is non-algebraic, and can only 
be calculated by iterative methods.          

Edwin