In this paper we present sufficient conditions under which a fairly large class of single-input non-linear systems including feedforward systems and the well-known ball-and-beam model, are globally asymptotically and locally exponentially stabilizable by smooth state feedback. A nested saturation controller with state-dependent saturation levels is constructed explicitly. using a novel design approach which combines the nested saturation strategy for marginally unstable linear systems subject to input saturation, with the small feedback design technique, developed for global asymptotic stabilization of general non-affine systems with marginally stable free dynamics. The power of the state-dependent saturation design method is demonstrated by solving a number of non-linear control problems, particularly, the global stabilization problem of a class of two-dimensional non-linear systems and the ball-and-beam system.