We observe condensation of exciton polaritons in quantum states composed of concentric rings when exciting cylindrical pillar GaAs/AlGaAs microcavities nonresonantly by a focused laser beam normally incident at the center of the pillar. The number of rings depends on the pumping intensity and the pillar size, andmay achieve 5 in the pillar of 40 μm diameter. Breaking the axial symmetry when moving the excitation spot away from the pillar center leads to transformation of the rings into a number of bright lobes corresponding to quantum states with nonzero angular momenta. The number of lobes, their shape, and location are dependent on the spot position.We describe the out-of-equilibrium condensation of polaritons in the states with different principal quantum numbers and angular momenta with a formalism based on Boltzmann-Gross-Pitaevskii equations accounting for repulsion of polaritons from the exciton reservoir formed at the excitation spot and their spatial confinement by the pillar boundary.