  We show that a charged, massive scalar field in the vicinity of an
electrically-charged Ay\'on-Beato-Garc\'ia (ABG) regular black hole has a
spectrum of quasibound states that (in a certain parameter regime) grow
exponentially with time, due to black hole superradiance. Superradiant
quasibound states are made possible by the enhancement of the electrostatic
potential at the horizon in nonlinear electrodynamics; in contrast, the
Reissner-Nordstr\"om black hole does not possess such superradiant quasibound
states. Here we compute the spectrum for a range of multipoles $\ell$ across
the parameter space, and we find the fastest growth rate in the monopole mode.
We find that a regular black hole with a small charge can still trigger a
significant superradiant instability if the charge-to-mass ratio of the field
is compensatingly large. We estimate the amount of black hole mass that can be
deposited in the scalar field, finding an upper bound of circa $20\%$ in the
extreme charge scenario. Finally, we consider the stationary bound states at
the superradiant threshold, and we conjecture that, due to this instability,
the ABG black hole will evolve towards a configuration with charged scalar
hair.
