Most incremental learners excessively prioritize object classes while neglecting various kinds of states (e.g. color and material) attached to the objects. As a result, they are limited in the ability to model state-object compositionality accurately. To remedy this limitation, we propose a novel task called Compositional Incremental Learning (composition-IL), which enables the model to recognize a variety of state-object compositions in an incremental learning fashion. Since the lack of suitable datasets, we re-organize two existing datasets and make them tailored for composition-IL. Then, we propose a prompt-based Composition Incremental Learner (CompILer), to overcome the ambiguous composition boundary. Specifically, we exploit multi-pool prompt learning, and ensure the inter-pool prompt discrepancy and intra-pool prompt diversity. Besides, we devise object-injected state prompting which injects object prompts to guide the selection of state prompts. Furthermore, we fuse the selected prompts by a generalized-mean strategy, to eliminate irrelevant information learned in the prompts. Extensive experiments on two datasets exhibit state-of-the-art performance achieved by CompILer.