The quest for a physical theory which describes gravitation as a quantum
phenomenon has been pursued under a wide variety of approaches, and with them, conceptions of the ultimate nature of time. We present one such approach, based upon causal sets, which conceives of the cosmos in terms of a quantum process of growth. In this conception, time arises as a consequence of unfolding of this growth process. The picture leads to a number of novel features, such as a description of a cosmos whose fundamental constants evolve under cycles of death and rebirth, an arrow of time which arises not from entropy of matter fields within the cosmos, but from entropy within the set of possible histories itself, and a sketch of why we might expect quantum gravity to give rise to a time varying cosmic acceleration. The description of time as growth may also provide insight into the dynamics of structures much smaller than the universe, such as the Internet or networks of trust among individuals.