The expensive brain hypothesis predicts an interspecific link between relative brain size and life-history pace. Indeed, animals with relatively large brains have reduced rates of growth and reproduction. However, they also have increased total lifespan. Here we show that the reduction
in production with increasing brain size is not fully compensated by the increase in lifespan. Consequently, the maximum rate of population increase (rmax) is negatively correlated with brain mass. This result is not due to a confounding effect of body size, indicating that the well-
known correlation between rmax and body size is driven by brain size, at least among homeothermic vertebrates. Thus, each lineage faces a "grey ceiling", i.e. a maximum viable brain size, beyond which rmax is so low that the risk of local or species extinction is very high. We found that
the steep decline in rmax with brain size is absent in taxa with allomaternal offspring provisioning, such as cooperatively breeding mammals and most altricial birds. These taxa thus do not face a lineage-specific grey ceiling, which explains the far greater number of independent origins of large brain size in birds than mammals. We also predict that (absolute and relative) brain size is an important predictor of macroevolutionary extinction patterns.