This paper proposes sufficient conditions under which nonlinear input-affine systems can be made Strongly iISS in the presence of actuator saturation. Strong iISS was recently proposed as a compromise between the strength of input-to-state-stability (ISS) and the generality of integral input-to-state stability (iISS). It ensures in particular that solutions are bounded provided that the disturbance magnitude is below a certain threshold, and that they tend to the origin in response to any vanishing disturbance. We propose a growth rate condition under which the bounded feedback law proposed by Lin and Sontag for disturbance-free nonlinear systems ensures Strong iISS in the presence of perturbations. We illustrate our findings on the angular velocity control of a spacecraft with limited-power thrusters. In the specific case of linear time-invariant systems with neutrally stable internal dynamics, we provide a simple static state feedback that ensures Strong iISS in presence of actuator saturations. This second result is illustrated by the robust stabilization of the harmonic oscillator. In both cases, we provide an estimate of the maximum disturbance amplitude that can be tolerated without compromising solutions' boundedness. (C) 2016 Elsevier Ltd. All rights reserved.