This project aims to motivate research in competitive human-robot interaction by creating a robot competitor that can challenge human users in certain scenarios such as physical exercise and games. With this goal in mind, we introduce the Fencing Game, a human-robot competition used to evaluate both the capabilities of the robot competitor and user experience. We develop the robot competitor through iterative multi-agent reinforcement learning and show that it can perform well against human competitors. Our user study additionally found that our system was able to continuously create challenging and enjoyable interactions that significantly increased human subjects' heart rates. The majority of human subjects considered the system to be entertaining and desirable for improving the quality of their exercise.