Ways to re-write/simplify the logarithm of a logartihm etc, for example $\log(\log(\log(\log(x))))$? I have been unable to find an answer to how to simplify or rewrite the expression when taking multiple logarithms, i.e. is there a more compact way to write $\log(\log(x))$ or for that matter $\log(\log(\log(\log(x))))$, while still using logarithms in the more compact expression?
And is there a formula to generalize it for any case of an arbitrary number of logs, for example taking 5 $log$'s, i.e. $\log(\log(\log(\log(\log(x)))))$, we could rewrite this as ... ?
Please note I am not looking for the limit as the number of $\log$'s go to infinity, as is resolved in this question or in this one.
EDIT: I need this to be able to compute the $y$, in $y=\log(\log(\log(...\log(x))))$
 A: Let us consider the function
$$f(x) = \log^{(5)}x = \log(\log(\log(\log(\log(x))))).$$
This function takes real values when
$$x\ge e_3 = e^{e^e},$$
where
$$e_{_0} = 1,\quad e_{_{k+1}} = e^{e_k}.$$
Derivatives of $f(x)$ are
\begin{align}
&f'(x) = \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)\log^{(4)}(x)},\\
&\log f'(x) = - \log(x) - \log^{(2)}(x) - \log^{(3)}(x) - \log^{(4)}(x) - \log^{(5)}(x),\\
&f''(x) = -f'(x)\Bigg(\dfrac1x + \dfrac{1}{x\log(x)} + \dfrac{1}{x\log(x)\log^{(2)}(x)} + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)}\\
& + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)\log^{(4)}(x)}\Bigg),\\
&f'''(x) = -f''(x)\Bigg(\dfrac1x + \dfrac{1}{x\log(x)} + \dfrac{1}{x\log(x)\log^{(2)}(x)} + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)}\\
& + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)\log^{(4)}(x)}\Bigg)\\
&+f'(x)\Bigg(\dfrac1{x^2} + \dfrac{1}{x\log(x)}\left(\dfrac1x+\dfrac1{x\log x}\right)\\
& + \dfrac{1}{x\log(x)\log^{(2)}(x)}\left(\dfrac1x+\dfrac1{x\log x}+\dfrac{1}{x\log(x)\log^{(2)}(x)}\right)\\
& + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)}\\
&\times\left(\dfrac1x+\dfrac1{x\log x}+\dfrac{1}{x\log(x)\log^{(2)}(x)} + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)}\right)\\
& + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)\log^{(4)}(x)}\Bigg(\dfrac1x + \dfrac{1}{x\log(x)} + \dfrac{1}{x\log(x)\log^{(2)}(x)} + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)} + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)\log^{(4)}(x)}\Bigg)\Bigg),\\
&f'''(x) = \dfrac{f''^2(x)}{f'(x)} - f''(x)+f'(x)\Bigg(\dfrac1{x^2} + \dfrac{1}{x\log(x)}\left(\dfrac1x+\dfrac1{x\log x}\right)\\
& + \dfrac{1}{x\log(x)\log^{(2)}(x)}\left(\dfrac1x+\dfrac1{x\log x}+\dfrac{1}{x\log(x)\log^{(2)}(x)}\right)\\
& + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)}\\
&\times\left(\dfrac1x+\dfrac1{x\log x}+\dfrac{1}{x\log(x)\log^{(2)}(x)} + \dfrac{1}{x\log(x)\log^{(2)}(x)\log^{(3)}(x)}\right)\Bigg)\dots
\end{align}
The value and the derivatives in the point $e_5$ are
\begin{align}
&f(e_5) = 1,\\
&f'(e_5) = e^{-\left(e_4+e_3+e_2+e+1\right)},\\
&f''(e_5) = -e^{-2\left(e_4+e_3+e_2+e+1\right)}\left(e_4e_3e_2e+e_3e_2e+e_2e+e+1\right),\\
&f'''(e_5) =e^{-3\left(e_4+e_3+e_2+e+1\right)}\left(e_4e_3e_2e+e_3e_2e+e_2e+e+1\right)^2\\
&+e^{-2\left(e_4+e_3+e_2+e+1\right)}\left(e_4e_3e_2e+e_3e_2e+e_2e+e+1\right)\\
&e^{-\left(e_4+e_3+e_2+e+1\right)}\left(e^{-2e_4}+e^{-2\left(e_4+e_3\right)}(e_4+1)+e^{-2\left(e_4+e_3+e_2\right)}(e_4e_3+e_3+1)+e^{-2\left(e_4+e_3+e_2+2\right)}(e_4e_3e_2+e_3e_2+e_2+1)+\right)\dots\\
&f(x)=f(e_5)+f'(e_5)(x-e_5)+\dfrac1{2!}f"(e_5)(x-e_5)^2+\dfrac1{3!}f'''(e_5)(x-e_5)^3+\dots
\end{align}
The standard procedure considered allows us to construct a Taylor series at an arbitrary point. At the same time, the usefulness of this series raises great doubts.
More useful looks a function of the form
$$\log(1+\log(1+\log(1+\log(1+\log(1+x))))),$$
which is defined for $x\ge0.$
