Wargames are simulations of conflicts in which participants' decisions influence future events. While casual wargaming can be used for entertainment or socialization, serious wargaming is used by experts to explore strategic implications of decision-making and experiential learning. In this paper, we take the position that Artificial Intelligence (AI) systems, such as Language Models (LMs), are rapidly approaching human-expert capability for strategic planning -- and will one day surpass it. Military organizations have begun using LMs to provide insights into the consequences of real-world decisions during _open-ended wargames_ which use natural language to convey actions and outcomes. We argue the ability for AI systems to influence large-scale decisions motivates additional research into the safety, interpretability, and explainability of AI in open-ended wargames. To demonstrate, we conduct a scoping literature review with a curated selection of 100 unclassified studies on AI in wargames, and construct a novel ontology of open-endedness using the creativity afforded to players, adjudicators, and the novelty provided to observers. Drawing from this body of work, we distill a set of practical recommendations and critical safety considerations for deploying AI in open-ended wargames across common domains. We conclude by presenting the community with a set of high-impact open research challenges for future work.