Eternal inflation produces pocket universes with all physically allowed vacua and histories. Some of these pocket universes might contain a phase of slow-roll inflation, some might undergo cycles of cosmological evolution, and some might look like the Galilean genesis or other ``emergent'' universe scenarios. Which one of these types of universe we are most likely to inhabit depends on the measure we choose in order to regulate the infinities inherent in eternal inflation. We show that the current leading measure proposals---namely, the global light-cone cutoff and its local counterpart, the causal diamond measure---as well as closely related proposals, all predict that we should live in a pocket universe that starts out with a small Hubble rate, thus favoring emergent and cyclic models. Pocket universes which undergo cycles are further preferred, because they produce habitable conditions repeatedly inside each pocket.