Just one thing to remember before reading; this is my opinion so don't get annoyed if I put a game you like lower down the pecking order.
1. FIFA 13
FIFA has everything I want in a FIFA game, but it isn't and never will be perfect but it is getting closer to "real" Football every year. There is so much to do in the game, and a lot of ways you can play; variety in team play and single play. FUT is just amazing and I cannot stop playing it, likewise with Career Mode. Unfortunately, my focusing of plentiful time on these 2 modes limits my gameplay from the other features in FIFA so I need to go and give those a try sometime soon.
2. Skyrim
What a beautiful game. There would be a list of over a million of things you can do and achieve in this game. You are your own character, you play and fight how you want with the skills that you earn and can build on to make your character as gifted as they can be in the desired area. So much to explore, so much to find, there is almost no end to this game. Bethesda has definitely not upset me this time.
3. Far Cry 3
The main thing that wants me to keep playing this game is to find out what I will do next; what is going to happen in this crazy island adventure whilst trying to be killed by a brilliant psychopathic mind that is Vaas. You can venture, follow the main quest, go out and do whatever you want. Everything is so well spread out, there is never nothing to do. Most importantly, there is nothing you can't achieve in this game. Absolutely brilliant.
4. Minecraft
There is no game more open than Minecraft. What can't you do in this game? You can build whatever you want, seriously, anything, just install mods and you are set for a million years. The game is so fun, and where your creative side is shown. There are many times where the game requires to make smart procedures so you don't end up in a bad scenario which is what you certainly don't want. Multiplayer tops the game off. Honestly, anything you want to do, you can do it in Minecraft.
5. Battlefield 3
Firstly, this game makes CoD looks like a laughing stock. Here, you have to know the game, you have to have skill. BF3 requires teamwork, intelligence and general non-noobiness. Why I love the game so much is that everything is up for grabs; you enter the fight and there are multiple objectives and there is plenty of time on the clock to fulfill whatever you wish. You enter the fight and there is action, being quick-minded is a bonus that is found very helpful in this game and if you aren't the quickest thinker, you learn to be. All I ever wanted in an FPS is here, in BF3.
I know this is only a list of 5, but these are the games that really stand out for me. There are no amazingly addicting games out there so far apart from these 3.