According to the great well-known Euler’s formula:

$$e^{i\theta }=\cos \theta +i\sin \theta$$

Let $\theta =\pi /2$, then:

$$\begin{array}{rl}{e}^{i\pi /2}& =cos(\pi /2)+isin(\pi /2)\\ & =0+i\times 1\\ & =i\end{array}$$

Now, doing a substitution $i$ with $e^{i\pi /2}$:

$$i^i=(e^{i\pi /2}{)}^i=(e^{\pi /2}{)}^{i^2}=e^{-\pi /2}=0.207879\dots$$

How beautiful!