Autonomous construction of deep neural network (DNNs) is desired for data streams because it potentially offers two advantages: proper model's capacity and quick reaction to drift and shift. While the self-organizing mechanism of DNNs remains an open issue, this task is even more challenging to be developed for standard multi-layer DNNs than that using the different-depth structures, because the addition of a new layer results in information loss of previously trained knowledge. A Neural Network with Dynamically Evolved Capacity (NADINE) is proposed in this paper. NADINE features a fully open structure where its network structure, depth and width, can be automatically evolved from scratch in an online manner and without the use of problem-specific thresholds. NADINE is structured under a standard MLP architecture and the catastrophic forgetting issue during the hidden layer addition phase is resolved using the proposal of soft-forgetting and adaptive memory methods. The advantage of NADINE, namely elastic structure and online learning trait, is numerically validated using nine data stream classification and regression problems where it demonstrates performance improvement over prominent algorithms in all problems. In addition, it is capable of dealing with data stream regression and classification problems equally well.