  We consider a two dimensional quantum Hamiltonian separable in Cartesian
coordinates and allowing a fifth-order integral of motion. We impose the
superintegrablity condition and find all doubly exotic superintegrable
potentials (i.e potentials V (x, y) = V_1(x)+V_2(y) where neither V_1(x) nor
V_2(y) satisfy a linear ODE) allowing the existence of such an integral. All of
these potentials are found to have the Painlev\'e property. Most of them are
expressed in terms of known Painlev\'e transcendents or elliptic functions but
some may represent new higher order Painlev\'e transcendents.
