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How can I compute the values of $n$ for which the following expression exists?

$$\log_e(\log_e(\log_e(\log_e(\ldots\log_e(n))))$$

It is for instance apparent that when $n = e$, the second application of $\log_e$ is undefined.

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As $\ln x$ is apriori defined only for $x>0$, you need $$n> e^{e^{e^{e^{e^0}}}} $$ where the tower has as many $e$s as your expression has $\log_e$s

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You have the sequence $$ 0,1,e,e^e,e^{e^e},e^{e^{e^e}},\dotsc, $$ for which for the $n$th member (starting with $0$), you can apply $\log$ only $n$ times before you end up with $\log{0}$. The sequence diverges, so there is no point at which you can take as many $\log$s as you like.

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