There are several results on the stabilization or regulation of a chain of integrators in the presence of a constant or time-varying delay in the input. However, the existing results commonly assume that the delay is restricted to be bounded. In this paper, the regulation problem of a chain of integrators with an unbounded time-varying delay in the input is considered. We propose an output feedback controller with a time-varying gain-scaling factor which is determined based on the condition of time-varying rate of the delay. As a benefit over the existing results, the delay is now allowed to be unbounded under the proposed condition. Also, uncertain delays can be accommodated with the proposed controller. Moreover, our proposed control scheme exhibits robustness such that the considered system can be extended to include some uncertain feedforward nonlinearities.