We present a state observer for a class of nonlinear systems with variable measurement delays, based on a modular design. An array of elementary high-gain observers with exponential decaying dynamics of the error are arranged to compensate arbitrarily large delays. We derive the relationship between the error decay rate, the observer gain and the bound on the delay of each stage of composite observer. The method can be applied to multiple output functions with independent delays but in this paper we focus on the case of scalar output to make the presentation easier to follow. We discuss the assumptions on the delay functions, and their relationship to common problems in the area of networked control systems, such as buffering, transmission delays, packet losses.