Aftershocks often occur within cascades of triggered seismicity in which each generation of aftershocks triggers an additional generation, and so on. The rate of earthquakes in any particular generation follows Omori's law, going approximately as 1/t. This function decays rapidly, but is heavy-tailed, and aftershock sequences may persist for long times at a rate that is difficult to discriminate from background. It is likely that some apparently spontaneous earthquakes in the observational catalogue are orphaned aftershocks of long-past main shocks. To assess the relative proportion of orphaned aftershocks in the apparent background rate, I develop an extension of the ETAS model that explicitly includes the expected contribution of orphaned aftershocks to the apparent background rate. Applying this model to California, I find that the apparent background rate can be almost entirely attributed to orphaned aftershocks, depending on the assumed duration of an aftershock sequence. This implies an earthquake cascade with a branching ratio (the average number of directly triggered aftershocks per main shock) of nearly unity. In physical terms, this implies that very few earthquakes are completely isolated from the perturbing effects of other earthquakes within the fault system. Accounting for orphaned aftershocks in the ETAS model gives more accurate estimates of the true background rate, and more realistic expectations for long-term seismicity patterns.