I was reading this article on MathWorld: https://mathworld.wolfram.com/DownArrowNotation.html,
and I decided to check the statement
$\ln^{*}n$ is the number of times the natural logarithm must be iterated to obtain a value $\leq e$.
But shouldn't it be $\leq 1$ ? I tried a simple check with the 3rd tetration of $e$. For $e \downarrow \downarrow e^{e^e} = \ln^{*}e^{e^e}$:
Try $\space \ln e^{e^e} = e^e \ln e \space \neq e$.
Try $\space \ln\ln e^{e^e} = e \ln (e\ln e) = e$.
This is, however, only 2 iterations. Could anyone help me find the problem?