i to the power i

Idea. Powers of complex numbers are defined through the exponential: $z^w=e^{w\log z}$. In exponential form, $i$ is easy to raise to any power.

Step 1. $i=e^{i\pi /2}$, so the principal logarithm of $i$ is $\log i=\frac{i\pi }{2}$.

Step 2. $$i^i=e^{i\cdot i\pi /2}=e^{-\pi /2}\approx \mathrm{0.208.}$$

A real number.

Other branches. Since $i=e^{i(\pi /2+2\pi n)}$ for every integer $n$, the other values of $i^i$ are $e^{-\pi /2-2\pi n}$, all real. The principal branch picks $n=0$.