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?