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We use the Aquarius simulations to show that the most massive subhalos in galaxy-mass dark matter halos in LCDM are grossly inconsistent with the dynamics of the brightest Milky Way dwarf spheroidal galaxies. While the best-fitting hosts of the dwarf spheroidals all have 12 < Vmax < 25 km/s, LCDM simulations predict at least ten subhalos with Vmax > 25 km/s. These subhalos are also among the most massive at earlier times, and significantly exceed the UV suppression mass back to z ~ 10. No LCDM-based model of the satellite population of the Milky Way explains this result. The problem lies in the satellites' densities: it is straightforward to match the observed Milky Way luminosity function, but doing so requires the dwarf spheroidals to have dark matter halos that are a factor of ~5 more massive than is observed. Independent of the difficulty in explaining the absence of these dense, massive subhalos, there is a basic tension between the derived properties of the bright Milky Way dwarf spheroidals and LCDM expectations. The inferred infall masses of these galaxies are all approximately equal and are much lower than standard LCDM predictions for systems with their luminosities. Consequently, their implied star formation efficiencies span over two orders of magnitude, from 0.2% to 20% of baryons converted into stars, in stark contrast with expectations gleaned from more massive galaxies. We explore possible solutions to these problems within the context of LCDM and find them to be unconvincing. In particular, we use controlled simulations to demonstrate that the small stellar masses of the bright dwarf spheroidals make supernova feedback an unlikely explanation for their low inferred densities.