First off, I would like to start by saying that there is no right or wrong answer. This thread is not an attempt to make a 'holier than thou' statement. But given how diverse games have become, and how different they all can be from one another, I attempted to try and understand what System Wars values the most in a video game. I hope that everyone will answer this question honestly.
If I had to rank it for myself, in decreasing order of importance, it would be Mechanics and Interactivity->Player Agency->Game Design->Sound Design->Controls->Graphics->Production Values->Story and Presentation->Replay Value and/or Lasting Appeal.
I do think all nine of them are variously important (depending on the kind of game it is- an epic story would make no sense in a Madden game, for example), but if I had to rank all of them, Mechanics and Interactivity would be first and foremost, with Player Agency a close second (for example, the reason I love Skyrim so much even though it has so many other failings is because it has so much Player Agency- I can do what I want, when I want, where I want; this is also the reason I disliked Skyward Sword, it took away too much Player Agency from me, especially compared to previous Zelda games).
Of course, often, I run into games I like a lot which seem to defy my preferences, but on the whole, I think the kinds of games I like are pretty consistent with my preferences up there.