Title: Online Learned Continual Compression with Stacked Quantization Modules. Abstract: We introduce and study the problem of Online Continual Compression, where one attempts to learn to compress and store a representative dataset from a non i.i.d data stream, while only observing each sample once. This problem is highly relevant for downstream online continual learning tasks, as well as standard learning methods under resource constrained data collection. We propose a new architecture which stacks Quantization Modules (SQM), consisting of a series of discrete autoencoders, each equipped with their own memory. Every added module is trained to reconstruct the latent space of the previous module using fewer bits, allowing the learned representation to become more compact as training progresses. This modularity has several advantages: 1) moderate compressions are quickly available early in training, which is crucial for remembering the early tasks, 2) as more data needs to be stored, earlier data becomes more compressed, freeing memory, 3) unlike previous methods, our approach does not require pretraining, even on challenging datasets. We show several potential applications of this method. We first replace the episodic memory used in Experience Replay with SQM, leading to significant gains on standard continual learning benchmarks using a fixed memory budget. We then apply our method to compressing larger images like those from Imagenet, and show that it is also effective with other modalities, such as LiDAR data.