While opportunities for spontaneous, undesigned, serendipitous science engagement (SSE) are abundant and evidence regarding its affordances is accumulating, little is known about its nature. In this paper, I present a model defining and identifying SSE, which consists of a personal and a contextual continuum. To explore the nature of family SSE, I analyzed a 66-minute case, obtained through self-ethnography, of three children and a mother engaging with bugs they discovered in their backyard. Employing the "language games" approach, the analysis elucidates how the absence of designated goals enabled legitimate multiple language-games over relatively long periods of time, with one game supporting the other. It also reveals delayed uptake, in which children appear to ignore scientific practice or content, while they are actually picking up ideas that they later use to engage with science. Examining processes of guided participation reveals that it entailed authentic modeling and gradual participation in the sense-making game, without abandoning other games. I discuss implications for science learning and for learning theory more broadly, including parents' role in SSE, issues of time for and in SSE, and guided participation in academic activities lacking designated goals.