While coherently driven Kerr microcavities have rapidly matured as a platform for frequency-comb formation, such microresonators generally possess weak Kerr coefficients; consequently, triggering comb generation requires millions of photons to be circulating inside the cavity. This suppresses the role of quantum fluctuations in the dynamics of the comb. In this paper, we realize a minimal version of coherently driven Kerr-mediated microwave-frequency combs in the circuit quantum electrodynamics (cQED) architecture, where the fluctuations of the quantum vacuum are the primary limitation on comb coherence. We achieve a comb phase coherence of up to 35 μs, approaching the theoretical device quantum limit of 55 μs and vastly longer than the inherent lifetimes of the modes, of 13 ns. The ability within cQED to engineer stronger nonlinearities than optical microresonators, together with operation at cryogenic temperatures, and the excellent agreement of comb dynamics with quantum theory indicates a promising platform for the study of complex dynamics of quantum nonlinear systems.