Base whose infinite power tower equals 2

Suppose $x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}=2$, an infinitely tall tower of exponents. What is $x$?

$x=\sqrt2$. The exponent on the bottom $x$ is the whole tower again, so $x^2=2$. One can check the tower $\sqrt2^{\sqrt2^{\cdots}}$ converges, and to $2$.