  We study a sample of 43 early-type galaxies, selected from the SDSS because
they appeared to have velocity dispersion > 350 km/s. High-resolution
photometry in the SDSS i passband using HRC-ACS on board the HST 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 sigma larger than 426 +- 30 km/s. These
objects define rather different 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
sigma". These objects appear to be of two distinct types: the less luminous
(M_r>-23) objects are rather flattened and extremely dense for their
luminosities -- their properties suggest some amount of rotational support and
merger histories with abnormally large amounts of gaseous dissipation. The more
luminous objects (M_r<-23) tend to be round and to lie in or at the centers of
clusters. Their properties are consistent with the hypothesis that they are
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 sigma 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
(M_r<-23.5), and BCGs, become slightly less round with increasing luminosity.
