This article addresses the stabilization problem for a class of stochastic feedforward systems with unknown nonlinearities. Due to the unknown nonlinearities, a nested saturation control law is designed with a priori unknown saturation bounds. Then, a group of logic-based switching conditions are provided for tuning the saturation bounds adaptively. Global asymptotical stability in the probability of the system states is proven. The key features of the control approach lie in that both explicit bounds and specific forms of the nonlinearities could be unknown for designing and deploying the controller.