We present Sequem, a sequential procedure that delivers point and confidence-interval (CI) estimators for extreme steady-state quantiles of a simulation-generated process. Because it is specified completely, Sequem can be implemented directly and applied automatically. The method is an extension of the Sequest procedure developed by Alexopoulos et al. in 2014 to estimate nonextreme steady-state quantiles. Sequem exploits a combination of batching, sectioning, and the maximum transformation technique to achieve the following: (i) reduction in point-estimator bias arising from the simulation’s initial condition or from inadequate simulation run length; and (ii) adjustment of the CI half-length to compensate for the effects of skewness or autocorrelation on intermediate quantile point estimators computed from nonoverlapping batches of observations. Sequem’s CIs are designed to satisfy user-specified requirements concerning coverage probability and absolute or relative precision. In an experimental evaluation based on seven processes selected to stress-test the procedure, Sequem exhibited uniformly good performance.