In this paper, we propose state predictors for stable genuinely nonlinear systems with time-varying measurement delay, with no restriction on its bound or serious limitations on the growth of the nonlinearities. The measurement delay is assumed to be continuous. A state prediction is generated by chains of nonlinear dynamic observers operating at different layers. On each layer, these observers reconstruct the unmeasurable state vector at different delayed time-instants, which partition the maximal variation interval of the time-varying delay. This partition determines the number of observers in the layer. Transitions from a layer to the next one are triggered by an online estimate of the magnitude of the state. Consistently, in passing to the next layer the partition is refined and the number of observers increased. In this sense, our predictor is nonlinear and adaptive.