What is the complexity of $i^i$?
This sequence is OEIS A173566. To understand how large it grows:
$a_n = 2^{2^{b_{n-1}}}$
where:
$b_0 = 0$
$b_n = b_{n-1} + 2^{b_{n-1}}$
The sequence $b_i$ grows faster than $2^{\cdotp^{\cdotp^{2^0}}}$, where there are $i$ 2's in the tower.
EXPTIME is $O(2^n)$, 2-EXPTIME is $O(2^{2^n})$, and in general, you can define n-EXPTIME . The sequence $b_i$ is not in n-EXPTIME for any natural n. So it, and therefore $a_i$, is not in the complexity class ELEMENTARY.
The definition above shows that $a_i$ is primitive recursive, which is interesting, because that means it doesn't grow as fast as the Ackermann function.
I think (but don't really have the time to formally prove or disprove right now) that means it's $\mathcal{E}^4$ in the Grzegorczyk hierarchy. Left as an exercise.