Only five populations have achieved maximum life expectancy (or best practice population) more than occasionally since 1900. The aim of this article is to understand how maximum life expectancy is achieved in the context of mortality transition. We explore this aim using the concepts of potential life expectancy, based on minimum rates at each age among all high longevity populations, and concordant ages. Concordant ages are defined as ages at which the minimum death rate occurs in the population with the maximum life expectancy. The results show the extent to which maximum life expectancy could increase through the realization of demonstrably achievable minimum rates. Concordant ages are concentrated at increasingly older ages over time, but they have produced more than half of the change in maximum life expectancy in almost all periods since 1900. This finding is attributed to their quantity and position whereby concordant ages are concentrated at the ages that have the greatest impact on mortality decline in a particular period. Based on mortality forecasts, we expect that concordant ages will continue to lead increases in female maximum life expectancy, but that they will play a weaker role in male maximum life expectancy.