The event rate, energy distribution and time-domain behaviour of repeating fast radio bursts (FRBs) contain essential information regarding their physical nature and central engine, which are as yet unknown1,2. As the first precisely localized source, FRB 121102 (refs. 3,4,5) has been extensively observed and shows non-Poisson clustering of bursts over time and a power-law energy distribution6,7,8. However, the extent of the energy distribution towards the fainter end was not known. Here we report the detection of 1,652 independent bursts with a peak burst rate of 122 h−1, in 59.5 hours spanning 47 days. A peak in the isotropic equivalent energy distribution is found to be approximately 4.8 × 1037 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 allows sensitive periodicity searches between 1 ms and 1,000 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, disfavouring emission mechanisms with large energy requirements or contrived triggering conditions.