A sleep deficit has far-reaching consequences, but for many people, healthy sleep is not a priority or a possibility. We explore the potential for "sleepy games" as a genre of transformational games. To explore this design space, we prototyped nine games through an iterative design process. Based on analysis of design decisions and the games as artifacts, we identify seven design challenges for sleepy games: agency and control; physiological and mental arousal; intervention timing; social embeddedness; multisensory experience; vulnerability; and identity and values. We expand on three games with playtesting to show how these design challenges unfold for players in practice, show the impact on players' lives, and discuss sleepy games as creative, social, and situated practices.