1. LoL/DOTA 2
These two games are extremely easy to pick up, only taking about 30 seconds to find a match, and can be extremely addicting. Being competitive games, there isn't any story to pay attention to and you can play the game based purely on visual cues.
2.FTL
All you need is a mouse for your treks through the vacuum of space, you don't need a monstrous rig to run it, so you can run it in windowed mode easily and have some videos going on the side. The game is really slow paced and relaxed so you can just sit back and upgrade your ship, take down pirates, and cruise through space. The game also resets after each death, bringing something new and refreshing with each play, but also keeping certain unlocks so your playtime doesn't feel completely useless.
3. Don't Starve/Minecraft/Other Sandbox Games
These kinds of games are perfect if you want to tune out the world and listen to some music. Just pop into your world, get creative, mine some supplies, kill some spiders or something, and feel pure bliss and adventure. Just be careful, if you don't keep an eye on the time you could end up losing two weeks of your life.
4.Civilization
The great thing about these games is that they take forever, so you can taken an entire afternoon to sit and watch some movies and take over the world, and you don't even need to worry about missing anything important in the movie because the game is turned based, so you can take a break and you won't miss anything. If the action really picks up in the movie, you can save on the exact turn you are on and pick it up at anytime. They also happen to be insanely fun.
5.World of Warcraft
Need to grind some dailies? Gotta get your crafting skills up? Farming your honor cap? All things that require little to no thought, which makes for a great game to play while watching your shows. Nothin' better than farming instance runs to level up while watching The Sopranos.
6.The Sims
The Sims is a game that's really easy to play for hours and hours. You can just sit and makes your Sims build skills, advance in careers, go to college, become supernatural creatures, and woohoo with every other sim in the neighborhood. The only speech is complete gibberish, so you don't have to actually listen to the game at all (even though it has some pretty funny sounds). Additionally, the Sims has it's very own pause button. You can pause the game at any point (or just let the normal time run cause the sims act pretty slow anyway) and click on new videos or watch important movie parts.
7. Binding of Isaac
The Binding of Isaac is a solid arcade type game that has tons of unlockables, secrets, and bosses for you to play with. As you progress through Mom's Basement you gain more and more powers making Isaac into an unstoppable little demon destroying all who dare to get in his way. If you happen to make it all the way through to the end, more end bosses unlock making you want to return, on top of randomly generated dungeons making each playthrough somewhat different. A really swell little game.