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So I like a good amount of different platforms of games so will try to break it down.
RPG's I go for story over just about everything else. That's what the genre is about, so if the story is shit then it defeats the purpose of the game. Shitty graphics can be made up with a good story (Xenogears has some pretty crappy graphics but it's one of my favorite games). One thing that a lot of people don't think about but will usually induce the nostalgia for a lot of RPG's is the music. Whenever I replay one of my favorites, it's usually because I thought about the music and remember how happy/excited/sad I was when that music played at that moment in the game. It's a context issue I'm sure but that's what I love. Interface is a small factor too that can turn me off to a game sometimes (as Pharaun mentioned, sometimes I just like to button mash). Legend of Dragoon was a great game but I hate the interface where you have to pay attention to grind. I love strategy RPG's like FFT and Tactics Ogre which increases replay for those kinds of games (always something different to try). Most traditional RPG's you only have one or two styles to play (magic or physical, for the most part) which increases replay to a certain point, but it does get old. I love trolling the Game FAQ boards for challenge games like for FFVII, the "Initial Equip, No Materia, Perfect Item" playthrough to keep the experience fresh.
Sports there's only football. I hate basketball and baseball games. Soccer is good but Football just piggybacks off my love of the game. Interface is key here. Crappy controls and bad options for plays can end it for me. Graphics are a good thing but my love started with Tecmobowl way back in the day so that's not really that big of a deal to me. Nowadays, there's all this manager customizations and trade beneficiaries and owners modes that just make me salivate over deep customization (like an RPG).
Fighting games I'm spoiled by Street Fighter and the original Mortal Kombat controls and basics. Tekken wasn't a good feel for me as it felt too realistic. As it (Fighting games as a whole) has evolved in to a more 3d fighting arena (as it is with newer MK games), it's lost it's 2D appeal and it's a bit more off of what made the games good. I love the new Street Fighter and the graphics and sounds are excellent but graphics never were good on the older games and the whole secrets section of these games make for incredible replay.
I'm gonna lump Survival Horror in to RPG's (though it has it's own genre now) but I love those types of games. Strategy and puzzle solving along with thrilling experiences and a good story make these games. I'm not exactly sure what genre Metal Gear is but these games are similar in their basic overall aspect (solve puzzles, kill baddies, get guns, win the game). Graphics/Music are king here. As Graphics get better, the experience gets better. Story has to be good and mind blowing twists and plots (ala Metal Gear) keep me interested.
Finally there's First Person Shooters. Started for me with Doom and eventually Halo and now Battlefield. Doesn't need a story to be good, just good controls, good gameplay and good action. Battlefield 3 is taking way too much of my time nowadays but it's captivating in the fact that here you have a beautiful game, tons of options for playstyle, tons of customization and regular updates, and the king of the experience? Online play. Literally an endless amount of challenges and foes just waiting to be played against and thousands of different strategies to evolve from playing against them.
With the introduction of online play to a lot of these games, I dare say it's pulled the human interaction movement back in to the spotlight. Of course you still have lone wolf mentalities and gamer fags (13 year olds) who will hate your guts for no reason but isn't it satisfying to slide a knife through their guts over and over again just to hear them whine and scream in to their headsets? Hope this helps (sorry for wall).
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The Defense is ready, Your Honor
Good, memorable music, IMO, though I place heavy focus on music because of being such a musically-inclined person. Even if a game wasn't made well, if the music was done well, I can still take a lot from it (or with me in the form of a MP3, of which I do even with classic chiptune songs).
We'll talk about fun, though. Fun is obviously subjective, but you know somewhere down the line of whether or not you're having it. Sometimes it is awarded during the journey of a game; other times upon accomplishing each milestone within a game. For some, the grind is fun, whether for good music, good combat, good pacing, a love of the characters, or just an obsession with being powerful or having "perfect" stats, and I think you can extend those feelings across a game as a whole.
Some people enjoy a game better when the narrative has valleys and peaks, such as an RPG's valley being the travel between two plot devices with random battles thrown in, and obviously, each peak being what transpires at those plot devices. Other games, like TF2 or a racing game, are far more liquid and dynamic, where players or the goings-ons of the current playthrough decide what constitutes a peak or a valley.
I suppose a good game either has good narriative or a good hook, keeps you relatively interested through what's going on within it, rewards you with whatever good things come out of it (whether a sweet item or a bitching boss theme/fight for making it somewhere, or lots of cash for doing something), and makes you want to play not far in the future, not for nostalga's sake alone, but because the game itself is "good".
Pardon me, I'm going to get back to playing FFIV, a good game.
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You wouldn't know that though because you've demonstrably never picked up a book nor educated yourself on the matter. Let me guess, overweight housewife?
gonna counter what some have said in regards to rpgs. I think on average I'm looking for a storyline that's going to keep me going, BUT if the gameplay is awesome and enjoyable the storyline can actually take a back seat (for me). Case in point, ffx-2 is one of my favorites of the ps FFs. The game play and side missions were so fucking fun; however, the storyline is pretty much crap. So it is duoable, though I think that's the only example I can come up with.