This paper investigates the problem of adaptive stabilization by state feedback for a class of high-order nonlinear systems with time-varying delays. Compared with the existing relevant literature, a distinguishing feature of the systems under investigation is that the growth rates are unknown functions of system states. This renders the existing control methods inapplicable to the control problems of the systems. By skillfully using the method of adding a power integrator, an adaptive state-feedback controller independent of the time delay is explicitly constructed with the help of appropriate Lyapunov–Krasovskii functionals. It is proved that the states of the time-delay nonlinear systems can be regulated to the origin, while all the closed loop signals are globally uniformly bounded. Finally, both physical and academic examples are provided to illustrate the effectiveness of the proposed scheme.