Climate sensitivity — the amount of global mean warming expected from a unit change in top-of-atmosphere radiative forcing — is the key diagnostic determining the severity of global warming, and the “fat tail” corresponding to very high climate sensitivity but low (or unquantifiable) a priori probability has deep implications for climate impacts since expected harms are typically dominated by fat-tail events. The standard analysis of climate sensitivity is based on linearisation of the energy balance equation, but in fact as (linearised) climate sensitivity increases, the next order nonlinear terms assume increasing importance. We discuss several implications of this elementary fact for the transient and equilibrium behaviour of climate in response to elevated CO2. First, it is pointed out that as linear climate sensitivity increases, the ability to detect the magnitude of climate sensitivity through observations of the Earth’s climate trajectory and its energy budget becomes increasingly difficult. We call this the Early Warning Problem. In the extreme, the Earth could be headed for a runaway greenhouse (we emphasise we do not think this at all likely), but the evidence for this trajectory would not become apparent for a century or more. Second, we point out that very high climate sensitivity affects the question of whether temperature continues to rise after net CO2 emissions are brought to zero. We will digress and discuss other factors that can cause committed warming after cessation of emissions of long-lived gases, notably those related to the land carbon cycle. Third, we will point out that when linearised climate sensitivity is high, the system is generically close to a bifurcation, but there is no advance warming of the “width” of the bifurcation; the bifurcation could be “small” (loss of arctic sea ice) or “large” (dissipation of low clouds, or even worse, transition to a runaway greenhouse). We will digress to a general discussion of the way low cloud feedbacks affect the bifurcation structure of the climate system. Finally, we will discuss the behaviour of stochastically forced climate-like systems that are near a bifurcation, and speculate on the relation of these results to behaviour of high-dimensional climate models.