Absorption and scattering are inherently related, as it is not possible to absorb power without creating a far-field shadow. We show, however, that properly overlapped resonant modes in a suitably designed system may in principle lead to arbitrarily large absorption levels, while at the same time minimizing the total scattering. We discuss the fundamental limits on scattering and absorption of an arbitrary receiving system and envision a composite nanoparticle that demonstrates the concept of a minimum-scattering superabsorber, with potential applications in energy harvesting, sensing, and imaging.