Abstract: Infinity is a profound concept, and hence a potent educational context for triggering mathematical thinking. Building on previous research concerning the ability of students to make sense of infinity, we ask how we could think about infinity as a mathematical playground. The infinity ‘equipment’ we have been designing for our mathematical ‘playground’ now constitutes a range of inquiry-oriented activities, all at varying stages along the design-experiment continuum. For this paper, we will consider activities built from Zeno’s paradoxes and Cantor’s cardinality of sets. Our work with children and young adults leads us to suggest a radical position: that infinity is indeed a playground for legitimate thinking, regardless of the content outcomes of that thinking.