There are many such classes of languages depending on the approach you want to use.  Let me focus on just one class: the **hyperarithmetic** (hyperarithmetical?) class, and try to explain why it is a very natural class, by mentioning several equivalent definitions of it:

* "*Computability in a type 2 functional*": a hyperarithmetic set of integers is one that can be computed by a "hyperarithmetical" machine, where a "hyperarithmetical" machine is one that can do everything that a Turing machine can do, but with the additional ability to decide whether a function $f\colon \mathbb{N} \to \mathbb{N}$ takes a nonzero value, where the function $f$ is *itself* computed by a hyperarithmetical machine.  (In other words, given a hyperarithmetical machine computing $f$, such that $f(n)$ is defined for every $n\in\mathbb{N}$, a hyperarithmetical machine can compute every $f(n)$ at once and decide whether there is one such that $f(n)\neq 0$, in which case of course it can also trivially find the corresponding $n$; if not every $f(n)$ is defined because the machine computing them does not halt, then the overall call will also not halt).  This definition is recursive, of course, and I didn't formalize it completely, but it still makes sense as the smallest which satisfies the conditions.  In the classical computability literature, this description is called "computability \[à la Kleene\] in the type 2 functional $\mathbf{E}$", but apparently it is never described in terms of "machines" like I tried to sketch.

* …In practice, a hyperarithmetical machine is one that can compute not only with integers, but also with exact real numbers (i.e., sequences of integers), not *all* real numbers but, precisely, those which are hyperarithmetical.  I think this makes the definition fairly natural.

* …Yet another way of rephrasing the same definition is that a hyperarithmetical machine is one that can compute infinite conjunctions/disjunctions (logical and/or), provided the terms of the conjunction/disjunction are themselves computed hypearithmetically.  This is just a rephrasing of the above, but it provides a link with certain kinds of infinitary logic.

* "*Metarecursion*": a set of integers is hyperarithmetic when it can be computed by a machine much like a register machine, except that the registers are allowed to hold ordinal values, the ordinals ranging up to the Church-Kleene ordinal (=smallest nonrecursive ordinal) $\omega_1^{\mathrm{CK}}$, and the machine is allowed to "loop" up to that value (I tried to summarize the idea of such ordinal computations [here](https://mathoverflow.net/questions/108949/various-definitions-of-recursion-from-ordinal-machines), in which terminology I would be speaking of $(\omega_1^{\mathrm{CK}},\omega_1^{\mathrm{CK}})$-machines).  These machines can compute much more than sets of integers, but those sets of integers which they can compute are precisely the hyperarithmetic ones.

* *The level $\Delta^1_1$ of the [analytical hierarchy](https://en.wikipedia.org/wiki/Analytical_hierarchy)* is again the class of hyperarithmetic sets: essentially those sets of integers which can be defined using one second-order quantifier, both in an existential and in a universal manner.  This is an analogue (the so-called "lightface" analogue) of one of the definitions of Borel sets in descriptive set theory.

* *Iterating the Turing jump*: a $0$-machine is just a Turing machine; a $0'$-machine is one that has access to an oracle that can tell it whether a $0$-machine halts; a $0''$-machine is one that has access to an oracle that can tell it whether a $0'$-machine halts; a $0^{(n)}$-machine is what you imagine; a $0^{(\omega)}$-machine (or arithmetical machine) is one that has access to an oracle that can tell it, given $n$, whether a $0^{(n)}$-machine halts; it is possible (although not completely trivial) to iterate this over the recursive ordinals, and hyperarithmetic sets are precisely those which are recognized by a $0^{(\alpha)}$-machine for some recursive ordinal $\alpha$.

* *The sets of integers belonging to the level $L_{\omega_1^{\mathrm{CK}}}$ of Gödel's [constructible universe](https://en.wikipedia.org/wiki/Constructible_universe)* where each level is defined essentially by adding every subset of the previous level that can be defined in it by a first-order formula.  Also, this level $L_{\omega_1^{\mathrm{CK}}}$ is the first one which satisfies a sizable amount of set theory, namely [Kripke-Platek](https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory).

Since all these definitions conspire to give the same class of hyperarithmetic sets, I think it's fair to say that it's a natural class.  There are plenty of classes both above and below, but I think this one deserves to be better known.

(Also, concerning complexity: there are also plenty of classes between $\mathsf{EXPTIME}$ and $\mathsf{REC}$: there are $\mathsf{ELEMENTARY}$ and $\mathsf{PR}$, but also lots of classes which can be defined between the class $\mathsf{PR}$ of primitive recursive sets/functions and that $\mathsf{REC}$ of recursive sets/functions, and that try to bridge the gap between complexity and computability.  These "subrecursive" hierarchies also deserve to be better known.)