In this paper, we present the first incremental parser for Tree Substitution Grammar (TSG). A TSG allows arbitrarily large syntactic fragments to be combined into complete trees; we show how constraints (including lexicalization) can be imposed on the shape of the TSG fragments to enable incremental processing. We propose an efficient Earley-based algorithm for incremental TSG parsing and report an F-score competitive with other incremental parsers. In addition to whole-sentence F-score, we also evaluate the partial trees that the parser constructs for sentence prefixes; partial trees play an important role in incremental interpretation, language modeling, and psycholinguistics. Unlike existing parsers, our incremental TSG parser can generate partial trees that include predictions about the upcoming words in a sentence. We show that it outperforms an n-gram model in predicting more than one upcoming word.