This paper investigates the problem of output-feedback adaptive stabilization control design for non-holonomic chained systems with strong non-linear drifts, including modelled nonlinear dynamics, unmodelled dynamics, and those modelled but with unknown parameters. An observer and an estimator are introduced for state and parameter estimates, respectively. By using the integrator backstepping approach and based on the observer and parameter estimator, a constructive design procedure for output-feedback adaptive stabilization control is given. It is shown that, under some conditions, the control design ensures the closed-loop system is globally asymptotically stable when there is no non-linear drift in the first subsystem, and semiglobally asymptotically stable, otherwise. An example is given to show the effectiveness of the theory.