I'm something of a hobbyist here, and would like to find a function that takes the natural log $e$ number of times so that I can define a new series of functions recursively, where $$f_0 = \ln(x), \quad f_1(x) = \frac{\ln^{\circledast e}(x)}{\ln (x)}, \quad \text{etc.}$$ wherein $$\ln^{\circledast e} \text{ or } \ln^{\circledast \text{number}}$$ is equivalent to \begin{equation*} \underbrace{\ln (\ln (\ln (\cdots)))}_{e} \text{ or } \underbrace{\ln (\ln (\ln (\cdots)))}_{\text{number}}. \end{equation*}
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1See Continuum between addition, multiplication and exponentiation?, especially the answer by Gottfried Helms. – Dave L. Renfro Jan 31 '23 at 20:29
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1See the search results for "functional square root". There is a general theory for such sequences $x_{n+1}=f(x_n)$ that proceeds by trying to find a power series expansion (usually around some fixed point) of a function $\phi$ with $\phi(f(x))=\phi(x)+1$ so that $f^n(x)=\phi^{-1}(\phi(x)+n)$. See https://math.stackexchange.com/questions/3373722/is-there-a-continuous-function-such-that-ffx-gx, mentioning a "classical Schröder mechanism" https://math.stackexchange.com/questions/2240712/functional-square-root-of-1-x1 – Lutz Lehmann Feb 01 '23 at 12:05