We present a mechanism that harnesses extremely weak Kerr-type nonlinearities in a single driven cavity to deterministically generate single-photon Fock states and more general photon-blockaded states. Our method is effective even for nonlinearities that are orders-of-magnitude smaller than photonic loss. It is also completely distinct from so-called unconventional photon blockade mechanisms, as the generated states are non-Gaussian, exhibit a sharp cutoff in their photon number distribution, and can be arbitrarily close to a single-photon Fock state. Our ideas require only standard linear and parametric drives and are hence compatible with a variety of different photonic platforms.