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Obviously, 111...=11^{1^{1^{.^{.^{.}}}}}=1111...=1
Furthermore, with a little bit of effort, we can show that 222...=2\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{.^{.^{.}}}}}=2222...=2
However, clearly 222...=∞2^{2^{2^{.^{.^{.}}}}}=\infty222...=∞
So what is the largest real number xxx such that xxx...x^{x^{x^{.^{.^{.}}}}}xxx... is a finite number?
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