Infinite Power Towers

Calculus Level 2

Obviously, $1^{1^{1^{.^{.^{.}}}}}=1$

Furthermore, with a little bit of effort, we can show that $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{.^{.^{.}}}}}=2$

However, clearly $2^{2^{2^{.^{.^{.}}}}}=\infty$

So what is the largest real number $x$ such that $x^{x^{x^{.^{.^{.}}}}}$ is a finite number?

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