Conversation is a cornerstone of social connection and is linked to well-being outcomes. Conversations vary widely in type with some portion generating complex, dynamic stories. One approach to studying how conversations unfold in time is through statistical patterns such as Heaps' law, which holds that vocabulary size scales with document length. Little work on Heaps' law has looked at conversation and considered how language features impact scaling. We measure Heaps' law for conversations recorded in two distinct mediums: 1. Strangers brought together on video chat and 2. Fictional characters in movies. We find that scaling of vocabulary size differs by parts of speech. We discuss these findings through behavioral and linguistic frameworks.