  We show that Franck-Condon physics leads to a significant current suppression
at low bias voltages (termed Franck-Condon blockade) in transport through
single molecules with strong coupling between electronic and vibrational
degrees of freedom. We find that transport in this regime is characterized by
remarkably large Fano factors ($10^2$--$10^3$), which arise due to
avalanche-like transport of electrons. Avalanches occur in a self-similar
manner over a wide range of time scales, as reflected in power-law dependences
of the current noise on frequency and vibrational relaxation rate.
