Finding all solutions to the equation $|||||x|-1|-1|-1|-1|=0$

Let $f(x) = ||x|-1|$. You have $f(x) = c$ iff $x=\pm(c+1)$ or $x=\pm (1-c)$.

The problem is to solve $f(f(f(f(x)))) = 0$.

This gives the possible values of $f(f(f(x)))$ as $\pm 1$.

This gives the possible values of $f(f(x))$ as $0, \pm 2$.

This gives the possible values of $f(x)$ as $\pm 1, \pm 3$.

Finally, this gives the possible values of $x$ as $0, \pm 2, \pm 4$.

The general pattern follows from this...

Your calculations look solid to me. Since it seems like you understand the computational aspects of this type of problem, I'd just like to offer a geometric perspective on the problem. Here's a graph of the function $f(x)=|||||x|-1|-1|-1|-1|$ for $-7<x<7$:

The five points where the function meets the $x$-axis correspond to the five roots of the equation you calculated. Note the periodic saw-tooth nature of the function between $-4$ and $4$. Once you know that $x=0$ is a root and $x=2$ is a root, you can automatically infer from symmetry that $x=-2,x=4,x=-4$ are also roots. All that's left at this point is logically justifying that these are the only roots.