Estimating the sizes of hard-to-count populations is a challenging and important problem that occurs frequently in social science, public health, and public policy. This problem is particularly pressing in HIV/AIDS research because estimates of the sizes of the most at-risk populations—illicit drug users, men who have sex with men, and sex workers—are needed for designing, evaluating, and funding programs to curb the spread of the disease. A promising new approach in this area is the network scale-up method, which uses information about the personal networks of respondents to make population size estimates. However, if the target population has low social visibility, as is likely to be the case in HIV/AIDS research, scale-up estimates will be too low. In this paper we develop a game-like activity that we call the game of contacts in order to estimate the social visibility of groups, and report results from a study of heavy drug users in Curitiba, Brazil (n = 294). The game produced estimates of social visibility that were consistent with qualitative expectations but of surprising magnitude. Further, a number of checks suggest that the data are high-quality. While motivated by the specific problem of population size estimation, our method could be used by researchers more broadly and adds to long-standing efforts to combine the richness of social network analysis with the power and scale of sample surveys.