Is it true that $f(x)^{f(x)^{\dots}}$ with $f(x) = -x^2 -x+1$ converges to the Golden Ratio? Why?

Question

My son accidentally discovered and, given $f(x)=-x^{2}-1x+1$, $f\left(x\right)^{f\left(x\right)^{f\left(x\right)^{f\left(x\right)}}}$ intersects 1.0 (i.e., $y=1.0$) at the negative of the golden ratio, i.e., $-1.618$ (and at the golden ratio -1 on the right side of the $y$ axis, i.e., $0.618$), and that adding EVEN numbers of repeated powers of $f(x)$ improves the precision (unsurprisingly). He discovered this just by playing around with graphs on desmos. I'd love to be able to explain to him why this is, and I'm pretty good at moderately advanced algebra, but I don't know what it means to raise a polynomial to the power of another polynomial! There's obviously something fairly simple going on, because by monkeying with the coefficients of the base polynomial you can get various versions of integer roots, and since the golden ration is basically a version of $\sqrt{5}$, I wasn't too surprised to see it there, but ... well, what does raising a polynomial to a polynomial power mean? Thanks!