In this note, a novel adaptive compensation control scheme is proposed for a class of non-linear systems with unknown control direction and a possibly infinite number of unknown actuator failures. By introducing a bound estimation approach, high-order Lyapunov functions and a Nussbaum function with faster growth rate, the obstacle caused by unknown failures and unknown control direction is successfully circumvented and all signals of the closed-loop system are proved to be globally uniformly bounded. Moreover, the proposed scheme is able to steer the tracking error into a predefined small residue set. Simulation results are presented to illustrate the effectiveness of the proposed scheme.