How would our world look like if the fine structure constant $\alpha$ were of order unity? While in our small $\alpha$ world an atom excited to the first excited state has negligible probability of decaying to the ground state while emitting more than a single photon, such processes are important in a large $\alpha$ world, making photon frequency conversion effective in the single-photon regime. Such behavior could be realized in superconducting circuit QED systems, where an artificial atom (qubit) is coupled to a Josephson junction array. The latter serves as a waveguide for microwave photons, but with Josephson instead of geometric inductance, allowing one to reach impedance of the order of the resistance quantum, which is equivalent to a high $\alpha$. We study a particular realization of this scenario, in which the artificial atom is a weak tunable Jospehson junction. We show that while the nonlinearity of the cosine potential only leads to weak photon inelastic scattering, phase slip processes can give rise to significant frequency conversion. Our results compare nicely with recent measurements by the Manucharyan group at Maryland.