We conceive the formation and spread of money in society using the metaphor of a game that actors learn to play, which is supported by the emergence of institutions that support this game. According to this viewpoint, Bitcoin is considered as an emergent money game formed by many groups and interests, competing with other established money games conducted in society. We discover that the Bitcoin money game has evolved through three distinct periods since its conception: a combative phase following Bitcoin's inception, a phase of horizontal integration with incumbent enterprises and merchants, and a time of rising vertical integration with powerful state actors. While accusations about Bitcoin's hostility toward the financial system's incumbents are not wholly unfounded, private enterprises and government agencies have made important contributions to the formation of the Bitcoin money game even to this day. Contesting the notion of Bitcoin as solely a disruptive force, we give a re-reading of Bitcoin's history to date, pointing out that Bitcoin may wind up strengthening the position of the central money games in society whose dominance it was intended to oppose.