Repeating Power

First attempt: $x^{\left({x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}}\right)}=4$ implies that $x^4=4$ . This means that $x=\pm\sqrt{2}$ . However, if $x<0$ and $x^{x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}}$ is well-defined, then $x=-1$ . This means that $x\neq -\sqrt{2}$ and hence $x=\sqrt{2}$ .

Everything seems alright now.

Checking: Let $a_0=\sqrt{2}$ and $a_{n+1}=\left(\sqrt{2}\right)^{a_n}$ for all $n\ge 0$ . Then we see that $a_0 <2$ and a_1=\sqrt{2}^\sqrt{2}<\sqrt{2}^2=2 . We can show that $a_n<2$ for all $n$ (by induction or some other method). Note that if $x=\sqrt{2}$ , then $4= x^{x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}}=\lim_{n \to \infty}a_n<2$ , a contradiction!

In fact, one can check that if $x^{x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}}=a$ has solution for $x$ , then $a=-1$ or $0<a\le e \approx 2.718$ .