In current practice, many image processing tasks are done sequentially (e.g. denoising, dehazing, followed by semantic segmentation). In this paper, we propose a novel multi-task neural network architecture designed for combining sequential image processing tasks. We extend U-Net by additional decoding pathways for each individual task, and explore deep cascading of outputs and connectivity from one pathway to another. We demonstrate effectiveness of the proposed approach on denoising and semantic segmentation, as well as on progressive coarse-to-fine semantic segmentation, and achieve better performance than multiple individual or jointly-trained networks, with lower number of trainable parameters.