Life is constructed from a limited toolkit: the Periodic Table. The reduction of oxygen provides the largest free energy release per electron transfer, except for the reduction of fluorine and chlorine. However, the bonding of O2 ensures that it is sufficiently stable to accumulate in a planetary atmosphere, whereas the more weakly bonded halogen gases are far too reactive ever to achieve significant abundance. Consequently, an atmosphere rich in O2 provides the largest feasible energy source. This universal uniqueness suggests that abundant O2 is necessary for the high-energy demands of complex life anywhere, i.e., for actively mobile organisms of approximately 10(-1)-10(0) m size scale with specialized, differentiated anatomy comparable to advanced metazoans. On Earth, aerobic metabolism provides about an order of magnitude more energy for a given intake of food than anaerobic metabolism. As a result, anaerobes do not grow beyond the complexity of uniseriate filaments of cells because of prohibitively low growth efficiencies in a food chain. The biomass cumulative number density, n, at a particular mass, m, scales as n (> m) proportional to m(-1) for aquatic aerobes, and we show that for anaerobes the predicted scaling is n proportional to m (-1.5), close to a growth-limited threshold. Even with aerobic metabolism, the partial pressure of atmospheric O2 (P(O2)) must exceed approximately 10(3) Pa to allow organisms that rely on O2 diffusion to evolve to a size approximately 10(3) m x P(O2) in the range approximately 10(3)-10(4) Pa is needed to exceed the threshold of approximately 10(2) m size for complex life with circulatory physiology. In terrestrial life, O(2) also facilitates hundreds of metabolic pathways, including those that make specialized structural molecules found only in animals. The time scale to reach P(O(2)) approximately 10(4) Pa, or "oxygenation time," was long on the Earth (approximately 3.9 billion years), within almost a factor of 2 of the Sun's main sequence lifetime. Consequently, we argue that the oxygenation time is likely to be a key rate-limiting step in the evolution of complex life on other habitable planets. The oxygenation time could preclude complex life on Earth-like planets orbiting short-lived stars that end their main sequence lives before planetary oxygenation takes place. Conversely, Earth-like planets orbiting long-lived stars are potentially favorable habitats for complex life.