We show that the simultaneous driving of a polariton condensate with both nonresonant and nth order resonant pump frequencies allows for a generic mechanism of breather formation. From this we construct for the second order resonance a family of exotic breathers with nontrivial discrete order of rotational symmetry. Finally, we demonstrate the spontaneous emergence of both crystalline and glassy orderings of lattices of polygonal breathers, depending on the degree of polygonal excitations at the lattice sites.