Story in every possible case.

I expect the graphics, gameplay, and story to fit the context. I do not expect FFXV graphics in every single game, especially on a mobile game. I do not expect FFVI level story in every game but I do expect a good mixture of graphics, story, and gameplay for the times we are living in to balance in a way that makes sense and is fun.

Cases in point:
Binding of Isaac, South Park, Terraria, Hyper Light Driffter - none of these have anything close to resembling great graphics, but they do have graphics that fit and they have great gameplay with just the right story.

I would say story. I actually feel graphics are one of the less important parts when it comes to a game, you don't need the best graphics when you have a good story (and gameplay, music, characters, etc...)

Story, if I were to just choose one of the two things presented without further details. But honestly a lot of different games on my all time favorite list does different things very well. Xenoblade told a great story, but lacked a lot of depth outside that and world size. I equally like Xeno X, because it provides so much depth and customization, and guves a real economic feel of trying to build civilization in video game form obviously. But story takes a hit. Different games offer different things. I guess a unique experience that grabs me is what I truly like most in games.

Very well said. I'd definitely agree with this view.

I suppose if I had to choose, I find a complelling story more appealing than great graphics, largely because the latter only applies in the era the game released in. Stories are timeless...graphics are not.

It should always be the STORY. Players are more into gaming if the storyline is exciting or reveals something extraordinaire. Graphics always come at second place as gamer wish to see beautiful environment it has. Also, players dwell more into the story which why they are hook to play it until the end of it. If the story runs so good, they might ask for season 2 or an update of the game to further improve what they've started. 

Gameplay > Story
and
Gameplay > Graphics,

but
whether Story > or < or = Graphics depends on the game.

A general rule can anyway be that Graphics must be at least "good enough" to make the game work well (for example in an FPS and in many RPGs and action games you'll be able to devise better tactics if you can spot and identify an enemy even from quite a long distance).

Both are important, Graphics can Augment a good story, help make it believable and even get you lost in the game world.
A good Story helps with that as well, it can make you fall in love with a character, hate another... It adds another dimension to making a world believable.

Depend, rpg, visual novel, point and click adventure > Story, action/hack and slash/beat em up/fighting/shooting/racing > graphic

There's a certain level of graphics a game needs to hit, though for me that level is no less than FFIX for polygonal, and no less than FFV for pixel.

After that, I get a diminishing return on how much wrote factor the graphics gives me. From that point on, story gets points more reliably. And graphics less than that will hamper the points given by a story.