Ground-based Doppler radar stations provide spatially complete precipitation data over much of the continental United States, probing spatial scales much smaller than the spacing between rain gauges. Here we determine the extreme-value behavior of the NEXRAD Stage IV radar-inclusive quantitative precipitation estimate (QPE; Lin & Mitchell 2005). In this QPE, the radar-precipitation relationship has been calibrated at each NEXRAD site using nearby rain gauges. We compare these extreme-value statistics to those of two other gridded QPEs which rely on rain gauge data only. We find good agreement between Stage IV and the Risser et al. (2019) technique, which fits the Generalized Extreme Value (GEV) distribution to the seasonal-maximum precipitation at each rain gauge and then interpolates the GEV parameters to produce a gridded map of extreme rainfall. The key distinction between this technique and most extant methods is that the extreme statistics are derived first, then spatially interpolated, as opposed to first spatially interpolating the precipitation, then deriving extreme statistics from the interpolated rainfall. The Risser et al. approach is demonstrably more accurate than the widely used operational Mountain Mapper algorithm, which significantly and systematically underestimates the magnitude of extreme rainfall everywhere east of the Rocky Mountains. These underestimates are caused by the algorithm's interpolation scheme, which "smooths over" extreme events by averaging spatially across many gauges. Our result underscores the importance of accounting for extremes in the creation and evaluation of gridded products, and points out potential systematic biases in commonly used QPEs. Applying a power spectrum analysis to the Stage IV extreme values, we find that the climatology of extremes in CONUS shows only slight variability on spatial scales smaller than ~100 km. Therefore, precipitation statistics derived from in-situ rain gauge data are of sufficient spatial resolution to faithfully capture extremes over most of the United States.