Infinite Power Towers

Calculus Level 2

Obviously, 111...=11^{1^{1^{.^{.^{.}}}}}=1

Furthermore, with a little bit of effort, we can show that 222...=2\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{.^{.^{.}}}}}=2

However, clearly 222...=2^{2^{2^{.^{.^{.}}}}}=\infty

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

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