  A study is presented of superintegrable quantum systems in two-dimensional
Euclidean space $E_2$ allowing the separation of variables in Cartesian
coordinates. In addition to the Hamiltonian $H$ and the second order integral
of motion $X$, responsible for the separation of variables, they allow a third
integral that is a polynomial of order $N\, (N\geq3)$ in the components $p_1,
p_2$ of the linear momentum. We focus on doubly exotic potentials, i.e.
potentials $V(x, y) = V_1(x) + V_2(y)$ where neither $V_1(x)$ nor $V_2(y)$
satisfy any linear ordinary differential equation. We present two new infinite
families of superintegrable systems in $E_2$ with integrals of order $N$ for
which $V_1(x)$ and $V_2(y)$ are given by the solution of a nonlinear ODE that
passes the Painlev\'e test. This was verified for $3\leq N \leq 10$. We
conjecture that this will hold for any doubly exotic potential and for all $N$,
and that moreover the potentials will always actually have the Painlev\'e
property.
