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We consider a two dimensional quantum Hamiltonian separable in Cartesiancoordinates and allowing a fifth-order integral of motion. We impose thesuperintegrablity condition and find all doubly exotic superintegrablepotentials (i.e potentials V (x, y) = V_1(x)+V_2(y) where neither V_1(x) norV_2(y) satisfy a linear ODE) allowing the existence of such an integral. All ofthese potentials are found to have the Painlev\'e property. Most of them areexpressed in terms of known Painlev\'e transcendents or elliptic functions butsome may represent new higher order Painlev\'e transcendents.

Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials