The apparently unrelenting growth in the GDP-share of health spending (SHS) has been a perennial issue of policy concern. Does an equilibrium limit exist? The issue has been left open in recent dynamic models which take income growth and population aging as given. We view these variables as endogenously determined within an overlapping-generations, human-capital-based endogenous-growth model, where a representative parent makes all life-cycle consumption and investment decisions, and life and health protection are subject to diminishing returns. Our prototype model, allowing for both quantity and quality of life as desired goods, yields equilibrium upper bounds for SHS. Our calibrated simulations also account for observed trends in reproductive choices, population aging, life expectancy, and economic growth. The analysis offers new insights about factors that drive long-term trends in aging and health spending and establishes a direct relation between health investments at young age and the equilibrium, steady-state rate of economic growth.