Does infinite tetration of negative numbers converge for any value other than -1? Okay, so I know that for positive values, $^{\infty}x$ converges to $-\frac{W(-\ln x)}{\ln x}$  for $e^{-e}\le x \le e^{\frac1e}$. Above that, it diverges. For positive values less than $e^{-e}$, any attempt to evaluate the infinite tetration oscillates between two values. $^{\infty}0$ is undefined, but $^{\infty}(-1) = -1$. By way of experimentation, other negative values of  $^{\infty}x$ seem to quickly become nonconvergent complex values. Are there any other negative values for which infinite tetration converges? If not, is there a proof that no other vaues do?
 A: 
Are there any other negative values for which infinite tetration converges?

Not really. Without going into too much detail, the potential map for the infinite exponential is:
$$\phi(z)=\exp(z/\exp(z))$$
If you plug in the above the parametrized unit circle $\exp(i\cdot\theta)$ for  $0\le\theta\le 2\pi$, you'll get what is known as the Shell-Thron region boundary on the Complex plane. In Maple for example:

phi:=z->exp(z/exp(z));
complexplot(phi(exp(I*theta)), theta = 0 .. 2*Pi, scaling = constrained);

Shell (of Shell-sort fame) in his Ph.D. thesis proved that convergence occurs only for $c$ inside this region on the complex plane. Alternatively, if you don't want to check against this region on the complex plane, you can check that the inverse potential map (multiplier) sends you inside the unit circle (which is crucial for convergence). That is, given $c$, check that:
$$t=|\phi^{-1}(c)|=|-W(-\log(c))|< 1$$
where $W$ is the principal branch of the Lambert function.
Baker and Rippon later made the result stronger, by proving that convergence occurs only either if $t<1$ or $t=1$ and $t^n=1$.
You can see the Shell-Thron region here (red nephroid).
The value (-1) is sort of a rogue in the complex plane and is the only value outside this region for which the infinite exponential trivially converges to itself, since ${^n}(-1)=-1$ for all $n\in\mathbb{N}$.
