There are plenty of synonyms for the phrase "kept busy." Some include occupied, engaged, involved, absorbed, immersed, and preoccupied. Other synonyms may include flooded, overwhelmed, loaded, and swamped. The concept of being "kept busy" implies that one is spending their time on various tasks and activities. Being occupied or engaged suggests a similar idea, while being involved or immersed may convey a sense of being deeply invested in one's work. To say that one is preoccupied or overwhelmed might suggest a more stressful or pressing situation, while describing oneself as flooded or swamped suggests a high volume of work.