The adaptive control problem of a class of strict feedback single-input single-output (SISO) nonlinear systems with input saturation is investigated in this paper. Different from the existing results, the considered virtual control coefficients can be unknown and the upper bound information is also unknown. A novel adaptive asymptotic tracking control scheme is developed by introducing some well-defined smooth functions and the bounded estimation approach. A Nussbaum function is introduced to compensate for the influence caused by input saturation and unknown virtual control parameters. By using the lower bound information of the unknown gain and delicately constructing a specific composite Lyapunov function for the closed-loop system as well as several useful inequalities, the need for upper bound information is eliminated and the global stability and tracking performance in the presence of input saturation can be guaranteed. Finally, the effectiveness of the proposed adaptive algorithm is illustrated with a practical simulation example and a comparative simulation.