World of Warcraft. 12 years of playing and raid/guild leadership helped me learn how to play, not just play. How to:

• Theorycraft
• Research how to improve
• Maximize output and/or efficiency
• Take advantage of class synergies in games
• Understand the importance of area of effect, burst damage, sustained single-target damage, etc.
• Understand damage mitigation vs avoidance, and where each is valuable
• Play to my/my team's strengths, rather than simply doing what is "best"
• Better recognize trends in game mechanics to anticipate what may come
• Recognize the valuable portions of a game's user interface and maximize its visibility while avoiding clutter

I had learned portions of these things in other games, but my leadership role in WoW pushed me to truly understand many things that aren't a major focus in most games.