  We show that dissipationless LCDM simulations predict that the majority of
the most massive subhaloes of the Milky Way are too dense to host any of its
bright satellites (L_V > 10^5 L_sun). These dark subhaloes have circular
velocities at infall of 30-70 km/s and infall masses of [0.2-4] x 10^10 M_sun.
Unless the Milky Way is a statistical anomaly, this implies that galaxy
formation becomes effectively stochastic at these masses. This is in marked
contrast to the well-established monotonic relation between galaxy luminosity
and halo circular velocity (or halo mass) for more massive haloes. We show that
at least two (and typically four) of these massive dark subhaloes are expected
to produce a larger dark matter annihilation flux than Draco. It may be
possible to circumvent these conclusions if baryonic feedback in dwarf
satellites or different dark matter physics can reduce the central densities of
massive subhaloes by order unity on a scale of 0.3 - 1 kpc.
