To conceive the infinity of integers, one has to realize: (a) the unending possibility of increasing/decreasing numbers (potential infinity), (b) that the cardinality of the set of numbers is greater than that of any finite set (actual infinity), and (c) that the leap from a finite to an infinite set is itself infinite (immeasurable gap). Three experiments probed these understandings via competitive games and choice tasks accompanied by in-depth interviews. Participants were children 6 to 15 years old and adults. The results suggest that roughly from about age 8 on, children grasp potential and actual infinity. However, for several additional years their conception of actual infinity is incomplete because the immeasurable gap between a finite and an infinite set is not entirely internalized. Even many adolescents and adults fail to appreciate this gap. Distinguishing between number concepts and their names facilitates conceiving aspects of infinity. Educational implications of these findings are discussed. (Contains 3 footnotes, 3 tables, and 3 figures.)