I remember this question from 40 years ago, and I never found a solution on my own.
If $Y = X ^ {X ^ Y}$:
When I attempt to reduce this, it always involves both an $X$ and $Y$ on both LHS and RHS of the equation.
I'm assuming that either 1. This is done via complex numbers or 2. This cannot be easily solved (it reminds me of a discussion of $x^y=y^x$).
You can make $X$ the subject of the formula using the Lambert function.
Take logs and multiply by $Y$ \begin{eqnarray*} Y \ln Y = X^Y \ln X^Y. \end{eqnarray*} Let $u= \ln X^Y $ \begin{eqnarray*} Y \ln Y = u e^u \end{eqnarray*} Now recall the Lambert $W$ function is defined by $we^w=z$ gives $w=W(z)$. So we have \begin{eqnarray*} \ln X^Y = W(Y \ln Y) \\ X= e^{W(Y \ln Y)/Y}. \end{eqnarray*}