*This question is very close to [this old MSE question of mine](https://math.stackexchange.com/questions/2068874/decidable-theories-of-arithmetic-with-nontrivial-arithmetic-hierarchies), which is still unanswered.*

Is there an (ideally reasonably-natural!) expansion of the structure $(\mathbb{N};+)$ in a finite language whose first-order theory is computable but such that, for each $n$, there is an $n$-quantifier formula not defining the same relation as any $<n$-quantifier formula?

For example, adjoining exponentiation with base $2$ does not work since its quantifier hierarchy collapses (due to [Cherlin/Point](https://webusers.imj-prg.fr/~francoise.point/papiers/cherlin_point86.pdf), see [here](https://mathoverflow.net/a/432844/8133)). Indeed, all the decidability results I'm familiar with for theories of arithmetic rely on quantifier elimination after some finite expansion by definitions, and so won't help here.