= Solution
Using the <principal complex logarithm>, $\operatorname{Log}i=i\pi/2$ and $\operatorname{Log}(-e)=1+i\pi$. Thus the values and their <real parts> and <imaginary parts> are
$$
\begin{array}{c|c|c|c}
\text{expression}&\text{principal value}&\operatorname{Re}&\operatorname{Im}\\\hline
\sqrt i&(1+i)/\sqrt2&1/\sqrt2&1/\sqrt2\\
\ln(-e)&1+i\pi&1&\pi\\
2^i&e^{i\ln2}&\cos(\ln2)&\sin(\ln2)\\
(i^i)^i&-i&0&-1
\end{array}
$$
For the final row, first calculate $i^i=e^{-\pi/2}$ and only then raise this positive real number to the power $i$. Treating nested powers as ordinary real powers would ignore the branching of <complex exponentiation>.
If all branches are requested, $\sqrt i$ also has the negative of its principal value, and the <complex logarithm> of $-e$ is $1+i(\pi+2\pi k)$. Allowing nonprincipal logarithms of the positive base $2$ gives $e^{-2\pi k}e^{i\ln2}$. Finally, all values of $(i^i)^i$ are $-ie^{-2\pi m}$, for integers $k,m$ as appropriate. These alternatives do not alter the principal-value table.
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