  We introduce the general polynomial algebras characterizing a class of higher
order superintegrable systems that separate in Cartesian coordinates. The
construction relies on underlying polynomial Heisenberg algebras and their
defining higher order ladder operators. One feature of these algebras is that
they preserve by construction some aspects of the structure of the
$\mathfrak{gl}(n)$ Lie algebra. Among the classes of Hamiltonians arising in
this framework are various deformations of harmonic oscillator and singular
oscillator related to exceptional orthogonal polynomials and even Painlev\'e
and higher order Painlev\'e analogs. As an explicit example, we investigate a
new three-dimensional superintegrable system related to Hermite exceptional
orthogonal polynomials of type III. Among the main results is the determination
of the degeneracies of the model in terms of the finite-dimensional irreducible
representations of the polynomial algebra.
