As an ensemble algorithm, network boosting enjoys a powerful classification ability but suffers from the tedious and time-consuming training process. To tackle the problem, in this paper, a broad network gradient boosting system (BNGBS) is developed by integrating gradient boosting machine with broad networks, in which the classification loss caused by a base broad network is learned and eliminated by followed networks in a cascade manner. The proposed system is constructed as an additive model and can be easily optimized by a greedy strategy instead of the tedious back-propagation algorithm, resulting in a more efficient learning process. Meanwhile, triple incremental learning capabilities including the increment of feature nodes, increment of input samples, and increment of target classes are designed. The proposed system can be efficiently updated and expanded based on the current status instead of being entirely retrained when the demands for more feature nodes, input samples, and target classes are proposed. The node-increment ability allows to add more feature nodes into the built system if the current structures are not effective for learning. The sample-increment ability is developed to allow the model to keep learning from the coming batch data. The class-increment ability is used to tackle the issue that the coming batch data may contain unseen categories. In comparison with existing popular machine learning methods, comprehensive results based on eight benchmark datasets illustrate the effectiveness of the proposed broad network gradient boosting system for the classification task.