  The event rate, energy distribution, and time-domain behaviour of repeating
fast radio bursts (FRBs) contains essential information regarding their
physical nature and central engine, which are as yet unknown. As the first
precisely-localized source, FRB 121102 has been extensively observed and shows
non-Poisson clustering of bursts over time and a power-law energy distribution.
However, the extent of the energy distribution towards the fainter end was not
known. Here we report the detection of 1652 independent bursts with a peak
burst rate of 122~hr^{-1}, in 59.5 hours spanning 47 days. A peak in the
isotropic equivalent energy distribution is found to be ~4.8 x 10^{37} erg at
1.25~GHz, below which the detection of bursts is suppressed. The burst energy
distribution is bimodal, and well characterized by a combination of a
log-normal function and a generalized Cauchy function. The large number of
bursts in hour-long spans allow sensitive periodicity searches between 1 ms and
1000 s. The non-detection of any periodicity or quasi-periodicity poses
challenges for models involving a single rotating compact object. The high
burst rate also implies that FRBs must be generated with a high radiative
efficiency, disfavoring emission mechanisms with large energy requirements or
contrived triggering conditions.
