I have tried to use $\ln$, but couldn't solve:
\begin{equation} \ln x^{x^{x^{2017}}}=x^{x^{2017}}\ln x=\ln 2017. \end{equation}
I found that $x=\sqrt[2017]{2017}$ is a solution, and it is easy to check it. But how to find that solution without guessing and how to prove if it is the only solution?
$$x^{x^{x^{2017}}}=2017$$
Raise $x$ to the power of both sides:
$$x^{x^{x^{x^{2017}}}}=x^{2017}$$ Let $y=x^{2017}$. Then the equality becomes: $$x^{x^{x^{y}}}=y$$ Since this is in the form of the first equation, $y$ can equal 2017. Therefore: $$y=2017=x^{2017}$$ Which gives us our one real solution $2017^{\frac{1}{2017}}$. By the fundamental theorem of algebra, there are 2016 more complex solutions, however it will take you a while if you don't have a calculator. Hope this helps!