  Regular black holes (RBHs) -- geometries free from curvature singularities --
arise naturally in theories of non linear electrodynamics. Here we study the
absorption, and superradiant amplification, of a monochromatic planar wave in a
charged, massive scalar field impinging on the electrically-charged
Ay\'on-Beato-Garc\'ia (ABG) RBH. Comparisons are drawn with absorption and
superradiance for the Reissner-Nordstr\"om (RN) black hole in linear
electrodynamics. We find that, in a certain parameter regime, the ABG
absorption cross section is negative, due to superradiance, and moreover it is
unbounded from below as the momentum of the wave approaches zero; this
phenomenon of ``unbounded superradiance'' is absent in the RN case. We show how
the parameter space can be divided into regions, using the bounded/unbounded
and absorption/amplification boundaries. After introducing a high-frequency
approximation based on particle trajectories, we calculate the absorption cross
section numerically, via the partial-wave expansion, as function of wave
frequency, and we present a gallery of results. The cross section of the ABG
RBH is found to be larger (smaller) than in the RN case when the field charge
has the same (opposite) sign as the black hole charge. We show that it is
possible to find ``mimics'': situations in which the cross sections of both
black holes are very similar. We conclude with a discussion of unbounded
superradiance, and superradiant instabilities.
