We can make it easier for disabled users to control assistive robots by mapping the user's low-dimensional joystick inputs to high-dimensional, complex actions. Prior works learn these mappings from human demonstrations: a non-disabled human either teleoperates or kinesthetically guides the robot arm through a variety of motions, and the robot learns to reproduce the demonstrated behaviors. But this framework is often impractical - disabled users will not always have access to external demonstrations! Here we instead learn diverse teleoperation mappings without either human demonstrations or pre-defined tasks. Under our unsupervised approach the robot first optimizes for object state entropy: i.e., the robot autonomously learns to push, pull, open, close, or otherwise change the state of nearby objects. We then embed these diverse, object-oriented behaviors into a latent space for real-time control: now pressing the joystick causes the robot to perform dexterous motions like pushing or opening. We experimentally show that - with a best-case human operator - our unsupervised approach actually outperforms the teleoperation mappings learned from human demonstrations, particularly if those demonstrations are noisy or imperfect. But our user study results were less clear-cut: although our approach enabled participants to complete tasks more quickly and with fewer changes of direction, users were confused when the unsupervised robot learned unexpected behaviors. See videos of the user study here: https://youtu.be/BkqHQjsUKDg