If
$W(z)$ denotes the solution to
$z = W\;e^W$,
$W \ge -1$, then for
$x = e^{W(\log(2))}$,
$x^{x^x}$ evaluates to
$2.4...$ for me. So maybe
$W$ denotes something else?
Also, "solve (x^(x^x))=2 for x > 0" gives the solution
$x = 1.47668$ on WolframAlpha, but it gives only a numerical solution:
https://www.wolframalpha.com/input?i=solve+%28x%5E%28x%5Ex%29%29%3D2+for+x+%3E+0
Changing "solve" to "plot" gives a graphical illustration confirming a solution near
$x=1.47...$:
https://www.wolframalpha.com/input?i=plot+%28x%5E%28x%5Ex%29%29%3D2+for+x+%3E+0
