Discrete time single species models with overcompensating density dependence and an Allee effect due to predator satiation and mating limitation are investigated. The models exhibit four behaviors: persistence for all initial population densities, bistability in which a population persists for intermediate initial densities and otherwise goes extinct, extinction for all initial densities, and essential extinction in which "almost every" initial density leads to extinction. For fast-growing populations, these models show populations can persist at high levels of predation even though lower levels of predation lead to essential extinction. Alternatively, increasing the predator's handling time, the population's carrying capacity, or the likelihood of mating success may lead to essential extinction. In each of these cases, the mechanism behind these disappearances are chaotic dynamics driving populations below a critical threshold determined by the Allee effect. These disappearances are proceeded by chaotic transients that are proven to be approximately exponentially distributed in length and highly sensitive to initial population densities.