We show that dissipationless Λ cold dark matter simulations predict that the majority of the most massive subhaloes of the Milky Way are too dense to host any of its bright satellites (LV > 105 L⊙). These dark subhaloes have peak circular velocities at infall of Vinfall= 30–70 km s−1 and infall masses of (0.2–4) × 1010 M⊙. 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.