Aiming at the problem of high-precision control of rotational speed in the rotary system of hydraulic anchor drilling rigs in the presence of parameter uncertainty and nonlinear constraints, a model-free adaptive iterative learning-based rotational speed control scheme for the rotary system of hydraulic anchor drilling rigs is proposed by taking advantage of the repetitive nature of the drilling rig operation. First, the state space model of the drill rig slewing control system about the rotational speed is constructed. Secondly, the dynamic linearization technique is used to construct the equivalent linear mapping relationship between the hydraulic motor and the servo valve current in the iterative domain of the drilling rig slewing system, and the model-free adaptive iterative learning speed control design method is proposed based on the historical servo valve current input and hydraulic motor rotary angle output data collected by the system. The asymptotic convergence of the rotational speed tracking error of the hydraulic anchor drilling rig slewing system along the data direction as well as in the direction of repeated operations is then given theoretically. Finally, the effectiveness of the algorithm is verified by joint simulation using MATLAB software and AMEsim platform. The results show that compared with the traditional PID algorithm and the iterative learning control algorithm, the proposed algorithm is able to realize the high-precision control of the drilling rig speed by using only the measurable data without the need of a known anchor drilling rig system model, and it still has a good adaptive and anti-jamming ability in the face of the sudden external disturbances and the fluctuation of the oil temperature.