We study a sample of 43 early-type galaxies, selected from the Sloan Digital Sky Survey (SDSS) because they appeared to have velocity dispersions of σ≥ 350 km s−1. High-resolution photometry in the SDSS i passband using the High-Resolution Channel of the Advanced Camera for Surveys on board the Hubble Space Telescope shows that just less than half of the sample is made up of superpositions of two or three galaxies, so the reported velocity dispersion is incorrect. The other half of the sample is made up of single objects with genuinely large velocity dispersions. None of these objects has σ larger than 426 ± 30 km s−1. These objects define rather different size–, mass– and density–luminosity relations than the bulk of the early-type galaxy population; for their luminosities, they are the smallest, most massive and densest galaxies in the Universe. Although the slopes of the scaling relations they define are rather different from those of the bulk of the population, they lie approximately parallel to those of the bulk at fixed σ. This suggests that these objects are simply the large-σ extremes of the early-type population; they are not otherwise unusual. These objects appear to be of two distinct types. The less luminous (Mr > −23) objects are rather flattened, and their properties suggest some amount of rotational support. While this may complicate interpretation of the SDSS velocity dispersion estimate, and hence estimates of their dynamical mass and density, we argue that these objects are extremely dense for their luminosities, suggesting merger histories with abnormally large amounts of gaseous dissipation. The more luminous objects (Mr < −23) tend to be round and to lie in or at the centres of clusters. Their circular isophotes, large velocity dispersions and environments are consistent with the hypothesis that they are brightest cluster galaxies (BCGs). Models in which BCGs form from predominantly radial mergers having little angular momentum predict that they should be prolate. If viewed along the major axis, such objects would appear to have abnormally large velocity dispersions for their sizes, and to be abnormally round for their luminosities. This is true of the objects in our sample, once we account for the fact that the most luminous galaxies (Mr < −23.5), and BCGs, become slightly less round with increasing luminosity. Thus, the shapes of the most luminous galaxies suggest that they formed from radial mergers, and the shapes of the most luminous objects in our large-σ sample suggest that they are the densest of these objects, viewed along the major axis.