We introduce Open-Universe Assistance Games (OU-AGs), a formal framework extending assistance games to LLM-based agents. Effective assistance requires reasoning over human preferences that are unbounded, underspecified, and evolving. Current LLM agents struggle in multi-turn interactions and with maintaining accurate models of user intent in collaborative settings. Existing assistance game formulations assume fixed, predefined preferences, an assumption that breaks down in open-ended dialogue where goals are revised incrementally and expressed in natural language. Grounded in cognitive science accounts of preference construction, we represent human preferences as a dynamically updated distribution over discrete natural-language goals. To operationalize OU-AGs, we introduce GOOD (GOals from Open-ended Dialogue), a data-efficient online method that extracts and ranks candidate goals during interaction, using LLM-simulated users to perform probabilistic inference over goal hypotheses. This allows for interpretable, uncertainty-aware preference representations without large offline datasets. We evaluate GOOD across three text-based domains: grocery shopping, household robotics (AI2-THOR), and coding. Compared to baselines without explicit goal tracking, GOOD produces semantically coherent goal representations and improves alignment with user intent across domains.