We introduce GuessingGame, a protocol for evaluating large language models (LLMs) as strategic question-askers in open-ended, opendomain settings.A Guesser LLM identifies a hidden object by posing free-form questions to an Oracle without predefined choices or candidate lists.To measure question quality, we propose two information gain (IG) metrics: a Bayesian method that tracks belief updates over semantic concepts using LLM-scored relevance, and an entropy-based method that filters candidates via ConceptNet.Both metrics are model-agnostic and support post hoc analysis.Across 858 games with multiple models and prompting strategies, higher IG strongly predicts efficiency: a one-standard-deviation IG increase reduces expected game length by 43%.Prompting constraints guided by IG, such as enforcing question diversity, enable weaker models to significantly improve performance.These results show that question-asking in LLMs is both measurable and improvable, and crucial for interactive reasoning.