Exponential growth in data generation and large-scale data science has created an unprecedented need for inexpensive, low-power, low-latency, high-density information storage. This need has motivated significant research into multi-level memory devices that are capable of storing multiple bits of information per device. The memory state of these devices is intrinsically analog. Furthermore, much of the data they will store, along with the subsequent operations on the majority of this data, are all intrinsically analog-valued. Ironically though, in the current storage paradigm, both the devices and data are quantized for use with digital systems and digital error-correcting codes. Here, we recast the storage problem as a communication problem. This then allows us to use ideas from analog coding and show, using phase change memory as a prototypical multi-level storage technology, that analog-valued emerging memory devices can achieve higher capacities when paired with analog codes. Further, we show that storing analog signals directly through joint coding can achieve low distortion with reduced coding complexity. Specifically, by jointly optimizing for signal statistics, device statistics, and a distortion metric, we demonstrate that single-symbol analog codings can perform comparably to digital codings with asymptotically large code lengths. These results show that end-to-end analog memory systems have the potential to not only reach higher storage capacities than discrete systems but also to significantly lower coding complexity, leading to faster and more energy efficient data storage.