I usually do not play games for the stories. The big exception for this I think is Metal Gear Solid 1. The game is alright but a bit stiff especially compared to other games in the series, and while the game I think has aged decently well, I always come back for the story. I just love how the events of the game unfolds and all the crazy codec conversations and twists. So I'm pretty open for a game to have an amazing story to make up for other aspects of the game.

As for games which really excel in gameplay, Mario 3 comes to mind. There isn't really much story to speak about and I don't think the original game (NES) graphics have aged the best, but the game itself is a blast. I've spent countless hours playing through all the levels over the years. So I think great gameplay can make up for a game lacking in story or graphics.

As for graphics, while I like a pretty game, I can't think of a single game I would play based on graphics alone. So I suppose that's the weakest link. Of course I think most games should aim to meet all 3 of those catergories or do 1/2 really, really well.

Gameplay > Story > Graphics

I have no problem with bad Graphics.
I can get by a average or even bad story, and i don't have a problem with the absence of one in some genres.
I can't stand bad gameplay.

Story = Gameplay = Graphics

Depends from game to game, it all just needs to come together nicely. Gameplay is obviously not a nr1 requirement to make a memorable game. (VNs, Heavy Rain, Journey, Walking Dead).

Graphics too are important, thats why they're called video games. I have enough of pixel art, 8 bit looking indies, save for a few good looking ones like Shovel Knight they all have garbage visuals.

It can very depending on the game. Not all my games need to be story driven, to be fun or a good experience. Not all games need to have great graphics, though they need to be good enough to not get in the way of the experience.
Game play is vital in the games with little to no stories. It's very important in games that don't have the bet graphics, and it needs to be at least neutral in games that are heavy on story and graphics. If it's getting in the way of the experience instead of being part of it or a good part of it, then the game wasn't made well.
I'd even accept bad game play in certain social situations where, the game is merely how people are interacting and not the reason people are interacting.

Gameplay above all, I can play games with great gameplay and non-existent story, but not the opposite in the vast majority of the cases (3DWorld is my favourite game this gen and it doesn't have almost any story).

I value great stories too, they can make me forget that the gameplay is bland, but it's not common. "To the moon" is one game I consider superb, but it barely has any gameplay at all.

I don't mind about graphics, I consider them to be a plus. I play Earthbound first time past year, and I don't mind the SNES graphics. It excels in every other aspect.

Remember last of us on ps3? That game have great story, great graphic, and great gameplay. Lets compare it to Mario galaxy on Wii. I prefer last of us

Gameplay > Story = Graphics

Gameplay is almost always what matters most. The story can be crap if that's the intention (like Mario), though if they try to have a good story and fail, that's a problem. Graphics are the same way. But if gameplay sucks, your game sucks.

The problem for me is I generally only enjoy a very specific type of gameplay. I liked platformers (mario) and action-puzzle (zelda) but it wasn't long that all those genres faded away to the likes of Final Fantasy 6, 7th saga, and other RPGs through the years.

To this day I can't really play anything other than RPGs/MMORPGs or more recently exploration/building games (minecraft/terraria).

So it's pretty much:

story > gameplay > graphics

But even if its got the most awesome gameplay and graphics ever, if its not an RPG/exploration type game i'm almost certain to not like it.