The magnitude-frequency distribution of boundary shear stress frames erosion rates in bedrock rivers, but empirical constraints are rare, particularly for extreme floods. Here we present measurements of mean stress τb and its frequency distribution along a fast-eroding river in Taiwan. We construct rating functions of discharge and hydraulic geometry using high-resolution satellite images of flood patterns, river stage time-series, topographic profiles, and post-flood field surveys. The method allows us to assess the spatiotemporal variation in τb along the channel. The boundary shear stress PDF p(τb) has a steep power-law tail, and includes semiannual floods generating τb ≈ 100–200 Pa and 50-year, 3000 m3 s−1 events driving τb ≥ 300 Pa. All such floods contribute to modification of the coarse alluvial cover and erosion of the bedrock bed. Given the steep tail decay in p(τb), the rapid channel incision probably owes more to the exceptional frequency of moderate shear stresses than to the magnitude of the extremes.