Analyzing large datasets requires responsive query execution, but executing SQL queries on massive datasets can be slow. This paper explores whether query execution can begin even before the user has finished typing, allowing results to appear almost instantly. We propose SpeQL, a system that leverages Large Language Models (LLMs) to predict likely queries based on the database schema, the user's past queries, and their incomplete query. Since exact query prediction is infeasible, SpeQL speculates on partial queries in two ways: 1) it predicts the query structure to compile and plan queries in advance, and 2) it precomputes smaller temporary tables that are much smaller than the original database, but are still predicted to contain all information necessary to answer the user's final query. Additionally, SpeQL continuously displays results for speculated queries and subqueries in real time, aiding exploratory analysis. A utility/user study showed that SpeQL improved task completion time, and participants reported that its speculative display of results helped them discover patterns in the data more quickly. In the study, SpeQL improves user's query latency by up to $289\times$ and kept the overhead reasonable, at $\$4$ per hour.