We present a first power-scaled nonlinear-mirror (NLM) modelocked thin-disk laser based on an Yb-doped gain material. The laser oscillator delivers average output powers up to 87 W and peak powers up to 14.7 MW with sub-600-femtosecond pulses at ≈9-MHz repetition rate. We demonstrate a threefold improvement in average output power and sixfold improvement in pulse energy compared to previous NLM-modelocking results. We obtain peak powers in excess of 10 MW for the first time from an NLM-modelocked laser oscillator. In our laser, the NLM is assisted by a semiconductor saturable absorber mirror (SESAM) to reliably initiate pulsed operation. We validate the high-power suitability of the NLM modelocking technique using low-absorption χ(2) crystals and optimized dichroic-mirror coating designs. Furthermore, we discuss stability against Q-switching and study how the tuning of the nonlinear mirror affects the laser performance.