Understanding how polyhedra pack into extended arrangements is integral to the design and discovery of crystalline materials at all length scales(1-3). Much progress has been made in enumerating and characterizing the packing of polyhedral shapes(4-6), and the self-assembly of polyhedral nanocrystals into ordered superstructures(7-9). However, directing the self-assembly of polyhedral nanocrystals into densest packings requires precise control of particle shape(10), polydispersity(11), interactions and driving forces(12). Here we show with experiment and computer simulation that a range of nanoscale Ag polyhedra can self-assemble into their conjectured densest packings(6). When passivated with adsorbing polymer, the polyhedra behave as quasi-hard particles and assemble into millimetre-sized three-dimensional supercrystals by sedimentation. We also show, by inducing depletion attraction through excess polymer in solution, that octahedra form an exotic superstructure with complex helical motifs rather than the densest Minkowski lattice(13). Such large-scale Ag supercrystals may facilitate the design of scalable three-dimensional plasmonic metamaterials for sensing(14,15), nanophotonics(16) and photocatalysis(17).