can't do anything else but agree

Is the gameplay good and fun to play, if yes you've got a good game. Level design and structure, too is an important factor.

Then I would say music. I'm a big fan of music and it adds a lot to a game. Whether its a sense of immersion or adding more impact to a scene.

Story follows. Though, I wouldn't consider THAT important unless it is a story driven game then it really depends on what the game is going for. If you don't meet criteria one though there is no point in playing a video game.

Graphics last. Most of my favourite games don't even have the best or realistic graphics, like Max Payne, SM World, TLOZ:LLTP for example. Its nice to have good graphics and can add to the experience but I would count this as bottom of my criteria.

If a game tries to maximize all the points above without impacting gameplay then you have ingredients of a great game. Good gameplay, fun, pretty to look at, memorable music, story and characters.

Gameplay is the core of it all and everything else has to serve gameplay, because graphics, sound and a story or premise are just as important as long as they serve the gameplay.

What makes some of the best fighting games for example, like Third Strike or Mark of the Wolves, is not only their great mechanics and flow of gameplay, but that the sound design and visuals work alongside the games, they wouldn't be as satisfying without their incredible sprites and appealing character design, fluid and to the point animations, the crunchy sound effects, etc. 

The balance of visuals come from how busy your detail is vs when you can tell what you can interact with in the environment, graphics have to be clear for the player above any amount of detail. Also comes into play stuff like dragged animations, stuff that goes on so long it becomes repetitive and frustrating, you can have a satisfying animation while still remaining short to serve gameplay.

Music can enhance atmosphere, the sense of fun and the like, there are games that I find well designed, but are not as entertaining to play because the music is so bad, bland and/or boring to listen to.

And a story or premise is more than its depth, pacing also comes into play, how long will you make a world in a simple platformer, when to drop the next boss in an action game. A simple story like Mario has a lot more charm depending on how that simple story unfolds, how Odyssey throws you into new creative worlds or a boss that you don't see coming, all is part of the pacing and telling a good story, no matter how simple or complex it is.