1. Gameplay
Barring a few exceptions, this is typically the most important aspect of a game for me. Why? Because it's a game. If I want top-notch music, I'll listen to an album. If I want top-notch writing, I'll read a book. If I want top-notch graphics, I'll watch a movie. I know that games make use of multiple mediums, but at their heart, they're still games. And a game with terrible gameplay, is failing at its most fundamental level.
2. Writing
Now to get into those "few exceptions" I was talking about. I cannot deny that I enjoy a good story with well-written characters. And if a game somehow manages to immerse me into its world despite the gameplay not being up to snuff, I can and usually will let it slide. For me, I usually end up coming for the gameplay, but staying for the story. But it needs to be the type of story I'm into, which is usually something cheesy. I'm not too fond of the "I couldn't make it in the film industry because I'm a terrible writer, so I'm making crappy games instead." type of stories you usually see. But that's just a personal preference, of which I'm clearly in the minority of if sales numbers are anything to go by.
3. Visuals
I'm ashamed to say, that I actually do like my games to look as pretty as possible. But even so, graphics are quite low in terms or priority for me, simply because they don't have a huge effect on the game. That being said, a game with a good art-direction can elevate a game from being mediocre to a masterpiece. Too bad art-direction has gone the way of the dodo in high budget games. ¬¬
I guess that's what happens when people keep telling the artists to make their art "pop" all the time. I never thought I'd find myself pining for the brown blurry bloom days, but here I am. Sure it was samey, but at least it was an attempt at art direction. But I digress.
4. Sound
I know that sounds is at the bottom of the list, but that doesn't mean it isn't important. A bad game with a good soundtrack can be more fun than a good game with a bad soundtrack. Put some heavy metal into your game, and chances are I'll think the game is better than it actually is, likewise if you put synth-wave in, and chances are I'll find the game worse than it actually is. And good use of sound effects is an often underappreciated aspect of games. But with all of that being said, it's not hard to mute a game and put on your own music. There's even the potential of having to play a game on mute simply to not disturb other people around you (perhaps you're playing it on a handheld). I had visuals low on the priority list because they don't have a huge effect on the game. Sound can quite possibly have zero effect on a game depending on said game, the person playing it, and the environment they're playing it in.
So that's my priorities. Although with that all being said, it's all quite relative.
I judge different games differently. With some games, I literally couldn't care less about the story, while with other games, the story is the only thing I care about. What I've written is just my general feelings on the matter.