Are there any more complex problems than the Totality Problem that appear in the practical world?
Let $\{W_e\}_{e\in\mathbb N}$ be a canonical list of the recursively enumerable, i.e., computably enumerable, i.e., Turing recognizable, languages (or sets).

The property that $W_e$ is actually recursive, i.e., computable, is $\Sigma^0_3$:
$$\exists d\forall n(n\in W_e\leftrightarrow n\not\in W_d).$$

It is at the 3rd level of the hierarchy: it is not of lower complexity than $\Sigma^0_3$. See e.g. Soare: Recursively enumerable sets and degrees, Theorem 3.4, for the proof.