"Co-production" is becoming an increasingly popular term in policymaking, governance, and research. While the shift from engagement and involvement to co-production in health care holds the promise of revolutionising health services and research, it is not always evident what counts as co-production: what is being produced, under what circumstances, and with what implications for participants. We discuss these questions and propose that co-production can be understood as an exploratory space and a generative process that leads to different, and sometimes unexpected, forms of knowledge, values, and social relations. By opening up this discussion, we hope to stimulate future debates on co-production as well as draw out ways of thinking differently about collaboration and participation in health care and research. Part of the title of this article is inspired by the book "The Social Construction of What?" by Ian Hacking (Cambridge, MA: Harvard University Press; 2000).