Help me visualize a function with infinitely many logarithms of absolute values of logarithms

To make sense of this, we might define $f_0(x)=x$, $f_{n+1}(x)=\ln(|f_n(x)|)$, and finally $f(x)=\lim_{n\to \infty}f_n(x)$. If this should converege for any $x$ at all then certainly to a number $y$ with the property $y=\ln(|y|)$, i.e., $e^y=|y|$. This has only one solution $\approx -0.5671432904$, so that $f$ has to be constant (where it converges). However, $f$ often fails to converge, for example if we reach $0$ after finitely many iterations.

Writing $g(x)=\log(|x|)$, you are looking at the iterates of the function $g$. The only possible limit, as Hagen von Eitzen points out, is the only fixed point of $g$, namely $a\approx-0.57$.

A fixed point is called stable if for $x$ close enough to $a$ iterating $g$ brings $x$ closer and closer to $a$. The stability of the fixed point at $a$ is related to the value of $|g'(a)|$; see Wikipedia. Since $|g'(a)|\approx1/0.57>1$, the fixed point is exponentially unstable: any point $x$ close to but not exactly at $a$ moves further away from it at exponential rate.

Therefore the only way a sequence $(x_i)$ given by $x_{i+1}=g(x)$ can converge to $a$ is that $x_N=a$ for some $N$. That is, you have to hit the limit exactly after a finite number of iterations. There are only countably many starting points $x_0\in\mathbb R$ that satisfy this; otherwise there is no limit.

The absolute value of the derivative of $x\mapsto\log(|x|)$ is $\approx1/0.57>1$ at the fixed point, so the fixed point is unstable. Therefore convergence to the limit is very rare even if you avoid zero: you have to hit the fixed point after finitely many iterations. It is possible to write the set of all such initial points recursively, but I'm not sure if that gives any additional insight.

Also only countably many initial points lead you to zero so that the sequence terminates and the definition doesn't make sense. For most real numbers (cocountably many) the iterates of $g$ are well defined but don't converge.

Conclusion: The limit function is only defined in countably many points and takes the same value $a$ at every such point.