Infinite Power Towers

Calculus Level 3

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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