ABSTRACT A handful of fast radio bursts (FRBs) are now known to repeat. However, the question remains – do they all? We report on an extensive observational campaign with the Australian Square Kilometre Array Pathfinder (ASKAP), Parkes, and Robert C. Byrd Green Bank Telescope, searching for repeat bursts from FRBs detected by the Commensal Real-time ASKAP Fast Transients survey. In 383.2 h of follow-up observations covering 27 FRBs initially detected as single bursts, only two repeat bursts from a single FRB, FRB 171019, were detected, which have been previously reported by Kumar et al. We use simulations of repeating FRBs that allow for clustering in burst arrival times to calculate new estimates for the repetition rate of FRB 171019, finding only slight evidence for incompatibility with the properties of FRB 121102. Our lack of repeat bursts from the remaining FRBs set limits on the model of all bursts being attributable to repeating FRBs. Assuming a reasonable range of repetition behaviour, at most 60 per cent (90 per cent confidence limit) of these FRBs have an intrinsic burst distribution similar to FRB 121102. This result is shown to be robust against different assumptions on the nature of repeating FRB behaviour, and indicates that if indeed all FRBs repeat, the majority must do so very rarely.