= Solution
The inner power takes the values $r_k=e^{-(\pi/2+2\pi k)}$. For each such real exponent, all values of the outer <complex exponentiation> are obtained from all <complex logarithm> values of its base $i$:
$$
i^{(i^i)}=\exp\left[i\left(\frac\pi2+2\pi m\right)e^{-(\pi/2+2\pi k)}\right],\qquad k,m\in\mathbb Z.
$$
Writing $\Theta_{k,m}=(\pi/2+2\pi m)e^{-(\pi/2+2\pi k)}$, the answer is
$$
\boxed{\Re(i^{(i^i)})=\cos\Theta_{k,m},\qquad\Im(i^{(i^i)})=\sin\Theta_{k,m},\qquad k,m\in\mathbb Z.}
$$
All values lie on the <unit circle>. The principal choice is $k=m=0$, giving $\cos[(\pi/2)e^{-\pi/2}]+i\sin[(\pi/2)e^{-\pi/2}]$. In particular the two nestings in parts (i) and (ii) are different: <complex exponentiation is not associative>.
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