Microblogging platforms such as Twitter have experienced a phenomenal growth of popularity in recent years, making them attractive platforms for research in diverse fields from computer science to sociology. However, most microblogging platforms impose strict access restrictions (e.g., API rate limits) that prevent scientists with limited resources-e.g., who cannot afford microblog-data-access subscriptions offered by GNIP et al.-to leverage the wealth of microblogs for analytics. For example, Twitter allows only 180 queries per 15 minutes, and its search API only returns tweets posted within the last week. In this paper, we consider a novel problem of estimating aggregate queries over microblogs, e.g., "how many users mentioned the word 'privacy' in 2013?". We propose novel solutions exploiting the user-timeline information that is publicly available in most microblogging platforms. Theoretical analysis and extensive real-world experiments over Twitter, Google+ and Tumblr confirm the effectiveness of our proposed techniques.