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We have witnessed in recent years an ever-growing volume of information becoming available in a streaming manner in various application areas. As a result, there is an emerging need for online learning methods that train predictive models on-the-fly. A series of open challenges, however, hinder their deployment in practice. These are, learning as data arrive in real-time one-by-one, learning from data with limited ground truth information, learning from nonstationary data, and learning from severely imbalanced data, while occupying a limited amount of memory for data storage. We propose the ActiSiamese algorithm, which addresses these challenges by combining online active learning, siamese networks, and a multi-queue memory. It develops a new density-based active learning strategy which considers similarity in the latent (rather than the input) space. We conduct an extensive study that compares the role of different active learning budgets and strategies, the performance with/without memory, the performance with/without ensembling, in both synthetic and real-world datasets, under different data nonstationarity characteristics and class imbalance levels. ActiSiamese outperforms baseline and state-of-the-art algorithms, and is effective under severe imbalance, even only when a fraction of the arriving instances' labels is available. We publicly release our code to the community.