I love games with a very well presented story (e.g. Splinter Cell 1, which makes you really believe you are this top agent, fighting against the bad guys). Graphics don't need to be the best of the best, blizzard shows that with every game( Diablo 2, Warcraft 3, Starcraft ).
On the other hand, if the game is addictive because of it's gameplay, it doesn't need a good story. For example the NHL series. I played 06-09 over years and its just a simple sportsgame.
Games which don't restrict you to a certain play type. Games such as battlefield 3 and 4 do this very well and there's quite a few maps to play which offer quick and slow gameplay, aswell as small maps with large maps, some games I feel like sitting back with a sniper on 1 point and defending it, other times I feel like playing operation metro and being in a large 32v32 battle for 1 point. Games such as RTS which are easy to learn, and hard to master, given that you'll want to stay if it's easy to learn, then once you've gotten into it, you can spend more time trying to perfect it.