Natural problems not in the Arithmetical Heirarchy What are some natural problems not in the Arithmetical Hierarchy?

The simplest example I know of is the theory of true arithmetic - that is, the set of all statements that are true of the natural numbers. This has degree $\mathbf{0}^{(\omega)}$, which computes every arithmetical set.
Noah Schweber above mentioned $WF$ and similar. My personal favorite is the theory of true second-order arithmetic, which is the set of true sentences about the set of subsets of the naturals. This is above $\Pi^1_n$ for every $n$, which makes it huge. It's also, incidentally, recursively isomorphic to the set of true sentences about the Turing degrees (think of "recursively isomorphic" as just "equivalent by way of a Turing machine", though it's a bit more than that).
Now, one thing worth noting is that everything I've suggested (and everything Noah had suggested as of the time of writing) is strictly above every arithmetic set. There are things that are off to the side, too - for example, there are Turing degrees that aren't arithmetic but compute no noncomputable sets (except themselves). But to my knowledge there are no natural examples of problems that are outside and not above the arithmetic hierarchy. I'd be interested in hearing if anyone has an example, though.
