This question follows from some interesting observations on a sum of reciprocals. Instead of summing them however, we will place each fraction to make a continued fraction.
Some visualisations on Desmos on nested functions have led to the following observations/questions. In particular, we will focus on nests that exhibit cycles, and hence a closed form for the functions in consideration can be deduced. Some issues arising from this will also be discussed.
Let us firstly start with the function $$f(x)=\frac1{1+\frac x{1+\frac x{1+\cdots}}}\implies \frac1{f(x)}=1+xf(x)\implies f(x)=\frac{-1+\sqrt{1+4x}}{2x}$$ where we take the positive root as $f(x)>0$ for $x>0$.
The domain is $x\in\left[-\frac14,\infty\right)\setminus\{0\}$, meaning that the closed form does not represent the values when $x<0$.
This is shown in the following plot; in red is $f$ represented as a nest, in green is $f$ in its closed form, and in blue is a function of the form $\tan(x^{1/k}\ln x)$, which is alike to $f$ for negative $x$ when reflected across the $y$-axis.
It is natural to think that solving $f(-x)=\frac1{1-\frac x{1-\frac x{\cdots}}}$ for itself suffices to tackle the problem, but note that this still brings up a rational function of a similar form.
Similarly, consider the function $$g(x)=\frac1{x-\frac1{x-\cdots}}=\frac1{x-g(x)}\implies g(x)=\frac{x\pm\sqrt{x^2-4}}2$$ where the domain now is $|x|\in[2,\infty)$. A simple inspection of the nest of $g$ (in here) reveals that there is much more to its behaviour outwith the domain. The function $g(x)$ is much 'denser' at $\pm2$ and displays more regular features after it.
Question
Is there a way to model the behaviour of such nested functions, for values that the domains of their closed forms do not cover?