The physics of earthquake triggering together with simple assumptions of self-similarity imply the existence of a minimum magnitude m(0) below which earthquakes do not trigger other earthquakes. Noting that the magnitude m(d) of completeness of a seismic catalog is not, in general, the same as the magnitude m(0) of the smallest triggering earthquake, we compare observed aftershock sequence parameters with the predictions made by the epidemic-type aftershock sequence model to constrain the value of m(0). In particular, we use quantitative fits to observed aftershock sequences from three previous studies, as well as Bath's law, to obtain four estimates of m(0). We show that the branching ratio n ( average number of triggered earthquakes per earthquake, also equal to the fraction of aftershocks in a seismic catalog) is the key parameter controlling the estimate of the minimum triggering magnitude m(0). Conversely, physical upper bounds for m(0) estimated from rate and state friction indicate that at the very least, 55% of all earthquakes are aftershocks.