Searching "Iterated logarithm" seems to give results only about the function $\log*(x)$. I would like to read about the properties of functions such as $$\begin{align} \log_{1}(x)&=\log(x) \\ \log_{2}(x)&=\log(\log(x)) \\ \log_{3}(x)&=\log(\log(\log(x))) \\ \log_{-1}(x)&=\exp(x)\\ \log_{0}(x)&=x \end{align}$$ etc. Are there any resources available on such functions? For example, their target series or limit definitions.
Thanks,
You can't go on iterating the $\log$ function like you mentioned in your question. This is because the $\log$ function is not defined for negative numbers$^\#$, and it decreases so fast that even if you seed your iteration with a reasonably large value of $x$, very soon, you'll get a negative number and run into illegal domain.
So, to make sure, your iterations are well defined, we introduce the $\log^*$ function which has the same inspiration as yours, but keeps my point in mind. That's why, you increase by $1$ every time you take $\log$ and iteratively define $$\log^*(x)=1+\log^*(\log(x))$$
$\#$ Logarithm of negative numbers may be handled using complex numbers. But, I don't know how to do that. Also, what I saud about the $\log^*$ function is the standard inspiration to define such a function. Maybe, you don't get much by including complex numbers, or else somebody would have done that by now :)