Digital games have traditionally been targeted at younger generations, although the proportion of older adult players is increasing. However, the design processes of digital games often do not consider the special needs of older adults. Co-design is a potential method to address this, but there is little research on co-designing games with older adults. In our study, we proposed a co-design process model that considers the intergenerational perspective. Using this model, eight older adults (two males and six females aged 47-80) and 22 sixth graders (11 males and 11 females aged 12-13) co-designed a digital game. The content of the game was based on old concepts used by the designers during their childhood. Similarly, game content involving new words and concepts were produced by the sixth graders. We collected data using semi-structured interviews and observations during the co-design process over a period of 24 months and then processed the data using grounded theory. The results indicated that the older adults identified seven game elements as essential to make games fun-appearance and aesthetics, competition, manageability of gameplay, social impact, familiarity, unpredictability, and intergenerational gameplay. Furthermore, we identified six assets that older adults have as game co-designers and five challenges that co-designing games with older adults may entail.