Recently, model predictive control (MPC) has attracted increasing attention due to the advantages of the clear concept, high dynamic performance and easy implementation. Conventional MPC for power converters is limited by sensitivity to electrical parameter mismatches, degrading performance in practical scenarios. To address the problem, this paper proposes an adaptive high-order extended state observer (HESO)-based model-free predictive control (MFPC) strategy for three-level neutral-point-clamped (3L-NPC) converters via a dual-component design. Specifically, the first part is a high-order extended state observer proposed to enhance disturbance estimation which improves estimation accuracy of system dynamics and lumped disturbances, outperforming conventional first-order observers. Moreover, the second part is an adaptive algorithm that utilizes the fast gradient method (FGM) under natural coordinate transformations, which decompose voltages and lumped disturbances into tangential and normal components to dynamically tune input gains for inductance mismatch compensation, thereby enabling nonparametric control. Finally, experimental results with inductance mismatch show the proposed approach improves current quality, and demonstrates better performance, when compared to conventional MPC and existing ESO-based methods.