We discuss the use of likelihood asymptotics for inference on risk measures in univariate extreme value problems, focusing on estimation of high quantiles and similar summaries of risk for uncertainty quantification. We study whether higher-order approximation, based on the tangent exponential model, can provide improved inferences. We conclude that inference based on maxima is generally robust to mild model misspecification and that profile likelihood-based confidence intervals will often be adequate, whereas inferences based on threshold exceedances can be badly biased but may be improved by higher-order methods, at least for moderate sample sizes. We use the methods to shed light on catastrophic rainfall in Venezuela, flooding in Venice, and the lifetimes of Italian semisupercentenarians.