The exponential growth in data generation and large-scale data analysis creates an unprecedented need for inexpensive, low-latency, and high-density information storage. This need has motivated significant research into multi-level memory systems that can store multiple bits of information per device. Although both the memory state of these devices and much of the data they store are intrinsically analog-valued, both are quantized for use with digital systems and discrete error correcting codes. Using phase change memory as a prototypical multi-level storage technology, we herein demonstrate that analog-valued devices can achieve higher capacities when paired with analog codes. Further, we find that storing analog signals directly through joint-coding can achieve low distortion with reduced coding complexity. By jointly optimizing for signal statistics, device statistics, and a distortion metric, finite-length analog encodings can perform comparable to digital systems with asymptotically infinite large encodings. These results show that end-to-end analog memory systems have not only the potential to reach higher storage capacities than discrete systems, but also to significantly lower coding complexity, leading to faster and more energy efficient storage.