Axions can be efficiently produced from the dynamical evolution of the electromagnetic fields in the polar caps of neutron stars. If the axion mass lies roughly in the range $10^{-10} \, {\rm eV} \lesssim m_a \lesssim 10^{-4}$ eV, a sizable fraction of the sourced axions will be gravitionally bound to the neutron star, accumulating on astrophysical timescales and producing high density clouds of axions. The characteristic density of these clouds near the neutron star surface can reach and exceed the local dark matter density by more than 20 orders of magnitude, producing environments in which the feeble nature of the axion coupling can be compensated for by large axion field values. We comment on a number of striking observational features that can arise in these systems, and demonstrate that existing radio telescopes can dramatically improve sensitivity to the axion-photon coupling.