To create and manipulate non-classical states of light for quantum information protocols, a strong, nonlinear interaction at the single-photon level is required. One approach to the generation of suitable interactions is to couple photons to atoms, as in the strong coupling regime of cavity quantum electrodynamic systems. In these systems, however, the quantum state of the light is only indirectly controlled by manipulating the atoms. A direct photon-photon interaction occurs in so-called Kerr media, which typically induce only weak nonlinearity at the cost of significant loss. So far, it has not been possible to reach the single-photon Kerr regime, in which the interaction strength between individual photons exceeds the loss rate. Here, using a three-dimensional circuit quantum electrodynamic architecture, we engineer an artificial Kerr medium that enters this regime and allows the observation of new quantum effects. We realize a gedanken experiment in which the collapse and revival of a coherent state can be observed. This time evolution is a consequence of the quantization of the light field in the cavity and the nonlinear interaction between individual photons. During the evolution, non-classical superpositions of coherent states (that is, multi-component 'Schrödinger cat' states) are formed. We visualize this evolution by measuring the Husimi Q function and confirm the non-classical properties of these transient states by cavity state tomography. The ability to create and manipulate superpositions of coherent states in such a high-quality-factor photon mode opens perspectives for combining the physics of continuous variables with superconducting circuits. The single-photon Kerr effect could be used in quantum non-demolition measurement of photons, single-photon generation, autonomous quantum feedback schemes and quantum logic operations.