Repeated logarithm By definition $\ln = \log_e$ on complex numbers is given by
$$
\ln(re^{i\theta}) = \ln(r) + i\theta
$$
$(-\pi < \theta\leq \pi, r >0)$. Then $\ln(-1) = \pi i$. And $\ln(\pi i) = \ln(\pi) + i\pi/2$.
If $\ln^{\circ n}(z) = \ln\circ\ln\circ \dots \circ\ln$ ($n$ times), is it possible to find what the exact value of 
$$
\lim_{n \to \infty} \ln^{\circ n}(-1)\quad 
$$
is?
From just using a calculator it seems like this actually converges. And from starting with for example $-2$ it looks like it converges to the same number. 
If it is not possible to find an exact value, how might one prove that this actually converges?
 A: The limit, if it exists, is a fixed point of $\ln$, and therefore of $\exp$.  Those fixed points are the branches of $-W(-1)$, where $W$ is the Lambert W function.  The values with imaginary part in $(-\pi, \pi]$ are the $0$ branch at $.3181315052-1.337235701 i$ and the $-1$ branch which is the complex conjugate of that.
The derivative of $\ln(z)$ is $1/z$.  Since $|1/z| < 1$ for each of these fixed points, they are both attracting, i.e. if some $\ln^{\circ n}(-1)$ is close enough to one of them, it will approach it in the limit.  In fact, if $p$ is one of these fixed points,
$$|p - \ln(z) | = \left|\int_\Gamma \frac{d\zeta}{\zeta}\right| \le  \frac{|z - p|}{|p| - |p-z|}$$ 
where $\Gamma$ is the straight line from $z$ to $p$.  So the basin of attraction includes
the open disk of radius $r$ around $p$ where $r/(|p|-r) = 1$, i.e. $r = |p|/2 = .6872785050$.  In this case $\ln^{\circ 3}(-1) =  .6645719224+.9410294873 i$ has distance approximately $.5263082050$ from $.3181315052+1.337235701 i$, and therefore future iterates will converge to that fixed point.
