depends on the game and genre, I am not one of those close minded people that only look for one or two things above all else in a game. I can enjoy different games for different reasons, a quality gaming experience can come from any combination of facets. There are games I have enjoyed for their story and characters, gameplay mechanics, content variety, on a technical appreciation level, on a visual level, novelty, for making me think or engaging on an emotional level,  how that game combines it's gameplay and level design into the story telling, long games, short games and everything in between.

What makes a quality game experience for me is a quality game experience no matter how the developers achieved that experience. 

Time to get properly immersed in the game without interruptions.

Nowadays that means shorter games that are faster to get into. My favorite games of all time are Everquest and Civilization, yet I wouldn't enjoy those today as I don't have regular 4 hour plus gaming sessions available anymore.

Other then that hard to say, depends on my mood. Variety is most important.

In my opinion, you've missed the most important aspect of video games.  The gameplay.  While there can be great games without having very good gameplay, and there has been, I feel gameplay should stand at the forefront of most every game.

Most games have storyline worse than the worst b rated movies, so with poor gameplay there's really no reason for me to bother with the game.  At the same time, while I'd like a good story, if the game has little to no story, or if it has a story but it's just poor, great gameplay will still make it a good game.  There's times where the story is way too intrusive and just ruins what good gameplay a game does have, but that's rare.. 

The worse the gameplay is, the shorter I'd like for the game to be.  Sure the game might have good gameplay, but is it good or fun enough to play for 30-40 hours before it starts to get stale?  This is where extras come in.  If the games gameplay can keep you entertained for a long period of time, then having a lot of extra content is a huge bonus.  I love playing games that have good extra content where I can continue playing the game long after I finished it. 

Finally there's graphics. I like having better graphics, but a bad game with good graphics will always be a bad game.

Some games like Just Cause 2 are the exception.  The story is horrible and the game has so many gameplay elements in the game that are done way better in other games, but Just Cause 2 is much more enjoyable than those same games with what you can do.  It's not just shooting, or driving a vehicle, or grappling everywhere, it's shooting while driving a vehicle only to open your parachute and then grapple to another vehicle and through whoever out of it.  It overall does a lot of things worse than other games, but the way you can easily just mix all this together makes it a great game.

I like to try new genres, so the criterias can change, but for what I call a great game, it needs :

1) A great scenario, and likeable characters. That's the scenario that makes me want to go further in the game.
2) A big replayability, and a large post game for grinding, and get some more infos on the characters etc
3) A lot of random, I like random. I can spend hundreds of hours on random dungeons.
4) Customizable characters is always a plus

For the questions :

Hours depends on the genre, I might say that 60 hours is ridiculous for some games

Visuals : I love nicely animated 2D sprites. But no indie pixel art crap, where it's almost not animated. And the HD and 60 fps is always a plus, if done correctly.

Extras : post game and customization and  procedural maps

Quality is such a bad word to use when talking about gaming.

Amount of content (hours) are pretty important to me. Many say they prefer short good games over long bad games, like it's a a sort of psychics law in this universe for a long game to be bad. Why not long and good games? Like Resident Evil 4, Bayonetta, Monster Hunter, Zelda, Dark Souls, Pikmin and many more.

Polish is also very important. It may sound weird, but when playing a game I want to notice as less as possible I'm playing a game. Good animations, no frequent (graphics)glitches and smooth controls.

The gameplay must be new in some sort form. It may borrow from other games, but also try to add new things. Recent Western games are very struggling with this. Almost every western game feels the same (controls, especially shooters I know the control scheme without even practicing or experimenting) and look the same because they try to be as realistic as possible. It's just boring. I personally hate it when big developers borrow engines from others; like Capcom using Cryengine for the new Monster Hunter Online. It does make your game look less unique and more lazy.

I don't care about story at all. Movies are for stories, games for gameplay. I prefer a game with a bad/boring story with actually good gameplay (Mario Galaxy) over a game with bad/boring gameplay and a good story (Bioshock Infinite). It is like many gamers are beginning to prefer the last one. Why do you want to 'struggle' through a game with boring or bad gameplay for the story? I don't know whats fun about that.

-Nice art style

-okayish story (should be okay for the genre Tetris does not need story Jump n Runs dont need MUCH story, racing games dont need story, Adventure and RPG DO need story)

-Long single player.

-Alot to explore (stuff to do)

-nice music

-smooth gameplay NO 1-25 FPS framerate.... (fuck this shit) constant 30/60 FPS

-should have almost or better NO bugs

-lot of little details that make a game stand out

- no immersion killing crap
(like STAGE CLEARED SILVER MEDAL or shit like that after every level of e.g Bayonetta.... I want to play with my own PACE and look around to enjoy the graphics etc I dont want to rush! There is a reason why I play games because I DONT WANT TO WORK! I can do this in the real world...

-no immersion killing crap 2
like quick time events or QE-boss kills like in GOW (Hello? I want to kill the boss I dont want to watch a movie of the boss being killed... Did I buy a movie or game?....) THEY SHOULD MAKE THIS SHIT OPTIONAL in the options!

I second this criteria!

With one addition - fun gameplay!

I just want to build off DevilRising's post, which really nails what's good and bad about video games past and present.

The beginning of any great video game, to me, is the mechanical structure at the center of the game, i.e., the set of moves and commands, the rules of the game, and the combination and variation of actions therein. How players experience the game via these mechanics, mentally and physically, is the most important thing. Secondary and tertiary criteria include graphics, art direction, music, sound, etc.

freedom controling character movement is very important if you ask me...everything else is fine I just dont like games where you feel stuck all the time..perfect example Tomb Raider on PS1 although that game is awesome lol

atmosphere and music for instance Metroid Prime, FF9, RE4, SM Galaxy..