This paper presents the design, deployment and qualitative study of a large-scale, public, generative art exhibition, through which passers-by could create artworks, and mint a non-fungible-token (NFT). Following the month-long exhibition, during which 229 anonymous participants produced artworks, 69 non-transferable NFTs were minted, we surveyed (33) and interviewed (14) expert and novice participants about their experiences. We explored contemporary challenges of owning digital things, and the extent to which NFTs, and 'Web3' technologies offer meaningful forms of ownership. Our findings describe how the inability to trade this NFT, and its unique circumstances of acquisition, made it meaningful in ways that extended beyond its immediate (limited) utility and offered participants something through which to construct identity. Reflecting on the aspirations, contradictions, and misconceptions of forms of ownership enabled by NFTs, we conclude with proposals for renewed attention in HCI to the nature of digital possessions, and the potential for 'ownership design'.@en