Long videos contain many repeating actions, events and shots. These repetitions are frequently given identical captions, which makes it difficult to retrieve the exact desired clip using a text search. In this paper, we formulate the problem of unique captioning: Given multiple clips with the same caption, we generate a new caption for each clip that uniquely identifies it. We propose Captioning by Discriminative Prompting (CDP), which predicts a property that can separate identically captioned clips, and use it to generate unique captions. We introduce two benchmarks for unique captioning, based on egocentric footage and timeloop movies - where repeating actions are common. We demonstrate that captions generated by CDP improve text-to-video R@1 by 15% for egocentric videos and 10% in timeloop movies.