Deep convolutional neural networks (DCNN) have enjoyed great successes in many signal processing applications because they can learn complex, non-linear causal relationships from input to output. In this light, DCNNs are well suited for the task of sequential prediction of multidimensional signals, such as images, and have the potential of improving the performance of traditional linear predictors. In this research we investigate how far DCNNs can push the envelop in terms of prediction precision. We propose, in a case study, a two-stage deep regression DCNN framework for nonlinear prediction of two-dimensional image signals. In the first-stage regression, the proposed deep prediction network (PredNet) takes the causal context as input and emits a prediction of the present pixel. Three PredNets are trained with the regression objectives of minimizing $\ell_1$, $\ell_2$ and $\ell_\infty$ norms of prediction residuals, respectively. The second-stage regression combines the outputs of the three PredNets to generate an even more precise and robust prediction. The proposed deep regression model is applied to lossless predictive image coding, and it outperforms the state-of-the-art linear predictors by appreciable margin.