This paper describes a hands-off socially assistive therapist robot that monitors, assists, encourages, and socially interacts with post-stroke users engaged in rehabilitation exercises. We investigate the role of the robot's personality in the hands-off therapy process, focusing on the relationship between the level of extroversion-introversion of the robot and the user. We also demonstrate a behavior adaptation system capable of adjusting its social interaction parameters (e.g., interaction distances/proxemics, speed, and vocal content) toward customized post-stroke rehabilitation therapy based on the user's personality traits and task performance. Three validation experiment sets are described. The first maps the user's extroversion-introversion personality dimension to a spectrum of robot therapy styles that range from challenging to nurturing. The second and the third experiments adjust the personality matching dynamically to adapt the robot's therapy styles based on user personality and performance. The reported results provide first evidence for user preference for personality matching in the assistive domain and demonstrate how the socially assistive robot's autonomous behavior adaptation to the user's personality can result in improved human task performance.