There are many synonyms for the phrase "keep someone busy." Some of the most common options include occupy, engage, involve, immerse, entertain, and captivate. Each of these words implies a sense of activity and involvement, suggesting that someone is actively doing something rather than simply passing the time. Other synonyms for the phrase might include stimulate, inspire, challenge, or even distract, depending on the specific context and desired effect. By using these alternate words to describe someone's activities, writers and speakers can add variety and depth to their language, allowing for a more nuanced and dynamic description of ongoing activities and engagements.