This article discusses approaches to determining the largest possible ground motion during an earthquake occurring near Earth’s surface. The problem is a difficult one because the available empirical data are insufficient to provide reliable indications of the upper tails of associated probability distributions. The largest ground-particle velocity ever measured during an earthquake is 3.18 m/sec. Some approaches yield limits that are roughly consistent with this number, but others suggest some potential to exceed it. We suggest that the highest particle velocity near the transition to elasticity indicates the maximum ground velocity possible in a shallow earthquake.