We demonstrate the formation of magnetically arrested minidisks (MAM) around equal-mass, nonspinning binary black holes with magnetohydrodynamic simulations of circumbinary disk accretion in full $3+1$ general relativity. The initial separation of $d\ensuremath{\sim}30M$ allows the black holes to host large minidisks that suppress the total rest-mass accretion rate variability, which is modulated primarily at $\ensuremath{\sim}1.6{f}_{\mathrm{orb}}$. Each black hole horizon saturates with dimensionless magnetic flux $\ensuremath{\phi}\ensuremath{\sim}30$. Magnetic reconnection near the horizons drives recurrent eruptions, which are expected to drive flaring in the infrared and x-ray bands. Our results establish MAMs as a new outcome of circumbinary disk accretion, and a promising source of novel electromagnetic counterparts to gravitational waves from binary black holes.