Conversations are intricately structured forms of social interaction in which talkers move through interconnected topics with nested levels of semantic specificity. What principles govern how conversational partners jointly navigate an expansive topic space? To characterize these dynamics, we introduce a new dataset of annotated topic shifts from N=1,505 annotators on 200 distinct video call conversations between strangers (Reece et al., 2023). Conversational dyads made stochastic but systematic transitions between topics, and within individual topics, we find that dyads begin concentrated in semantic space before dispersing to more idiosyncratic regions as topics progress. The same dispersion pattern also holds over entire conversations, providing quantitative evidence for nested levels of increasing specificity over conversations. Overall, our findings suggest that strangers get to know one another through systematic exploration of topic space, revealing hierarchical structure in idle talk.