Wind energy is one of the main renewable energy sources in the current energy transition. Due to ever more and ever larger wind turbines (WT), the requirements for WT operation become more complex. Model predictive control (MPC) for WTs shows the potential to handle these requirements and conflicting control objectives in a single optimization-based controller. Recent research has widely investigated MPC for WT in simulation, but mostly lacks experimental validation. This work aims to experimentally validate MPC on a full-scale WT under real conditions. To this end, we combine an Extendend Kalman Filter for nonlinear state estimation with robust linear time-varying MPC. We evaluate the proposed control algorithm in terms of time-domain performance and power curve in simulation. However, the main contribution of this work is the experimental validation on a 3MW WT in Northern Germany with a total duration of 3h continuous full access of the controller. We were able to demonstrate stable operation of the proposed MPC in the upper partial load regime, transition regime and lower full load regime, at measured wind speeds between 4.76m/s and 13.06m/s, inside and outside the wake shadow of another WT. The power curve determined in simulation shows comparable results to a reference feedback controller. The MPC formulation combines several control objectives in a single optimization problem, yet the tuning effort still remains complex. In future work, we plan to reduce the complexity of the control loop based on this experimentally validated MPC. We provide our experimental data at https://github.com/rwth-irt/MPC_WT_experiment.