The analysis of the demographic transition of the past century and a half, using both empirical data and mathematical models, has rendered a wealth of well-established facts, including the dramatic increases in life expectancy. Despite these insights, such analyses have also occasionally triggered debates which spill over many disciplines, from genetics, to biology, or demography. Perhaps the hottest discussion is happening around the question of maximum human lifespan, which –besides its fascinating historical and philosophical interest– poses urgent pragmatic warnings on a number of issues in public and private decision-making. In this paper, we add to the controversy some results which, based on purely statistical grounds, suggest that the maximum human lifespan is not fixed, or has not reached yet a plateau. Quite the contrary, analysis on reliable data for over 150 years in more than 20 industrialized countries point at a sustained increase in the maximum age at death. Furthermore, were this trend to continue, a limitless lifespan could be achieved by 2102. Finally, we quantify the dependence of increases in the maximum lifespan on socio-economic factors. Our analysis indicates that in some countries the observed rising patterns can only be sustained by progressively larger increases in GDP, setting the problem of longevity in a context of diminishing returns.