When two separate masses of the same fluid are brought gently into contact, they are expected to fully merge into a single larger mass to minimize surface energy. However, when a stationary drop coalesces with an underlying reservoir of identical fluid, merging does not always proceed to completion. Occasionally, a drop in the process of merging apparently defies surface tension by 'pinching off' before total coalescence occurs, leaving behind a smaller daughter droplet. Moreover, this process can repeat itself for subsequent generations of daughter droplets, resulting in a cascade of self-similar events. Such partial coalescence behaviour has implications for the dynamics of a variety of systems, including the droplets in clouds, ocean mist and airborne salt particles, emulsions, and the generation of vortices near an interface. Although it was first observed almost half a century ago, little is known about its precise mechanism. Here, we combine high-speed video imaging with numerical simulations to determine the conditions under which partial coalescence occurs, and to reveal a dynamic pinch-off mechanism. This mechanism is critically dependent on the ability of capillary waves to vertically stretch the drop by focusing energy on its summit.