Here are some games that I enjoyed playing quite a bit and were either released or updated in some form in 2013.
League of Legends
Far and away my personal game of the year was League of Legends. If you want to get specific to 2013, then “Season 3” of League of Legends. I truly identified as a League player during 2013 and it’s where the vast majority of my video game playing time was devoted. In addition to that I spent countless hours watching professional play and reading the forums and subreddit. I went from a new player to a Silver I rank and that profound, painful, hilarious, rewarding mess of an experience was something I’ll never forget. I’ve taken a break from the game since Season 3, but I hope to return shortly.