We show how to achieve fast autocompletion for SPARQL queries on very large knowledge bases. At any position in the body of a SPARQL query, the autocompletion suggests matching subjects, predicates, or objects. The suggestions are context-sensitive in the sense that they lead to a non-empty result and are ranked by their relevance to the part of the query already typed. The suggestions can be narrowed down by prefix search on the names and aliases of the desired subject, predicate, or object. All suggestions are themselves obtained via SPARQL queries, which we call autocompletion queries. For existing SPARQL engines, these queries are impractically slow on large knowledge bases. We present various algorithmic and engineering improvements of an existing SPARQL engine such that these autocompletion queries are executed efficiently. We provide an extensive evaluation of a variety of suggestion methods on three large knowledge bases, including Wikidata (6.9B triples). We explore the trade-off between the relevance of the suggestions and the processing time of the autocompletion queries. We compare our results with two widely used SPARQL engines, Virtuoso and Blazegraph. On Wikidata, we achieve fully sensitive suggestions with sub-second response times for over 90% of a large and diverse set of thousands of autocompletion queries. Materials for full reproducibility, an interactive evaluation web app, and a demo are available on: https://ad.informatik.uni-freiburg.de/publications .