Time to stop beating around the bush.
I've come to realize that although I've spent a good deal of the week exploring the various methods that games have used to convey story, I haven't really genuinely tackled the question upon which I delved into this week: "Do games need to tell a story"? Well the short answer is no, not really.
As I established in the first day, a game does need a premise: the foundation upon which the architecture is built is essential because without the core idea the entire thing becomes a rather aimless and ambling exercise that probably won't wind up as a game -- if it indeed winds up as much of anything -- at all. Once you have that premise though, then everything else is in the most basic terms just a gussying up. Consider that a game like Super Mario Brothers could just as easily be expressed in a game where a stick-man runs through mostly featureless plains and interacts with enemies that are rudimentary geometry like squares and circles. The game has the same core premise as Super Mario Brothers -- get to the end of the final level and win the game -- but with all the extra things like level design, art, fleshed out characters, and story stripped out.
Of course, games like that are rarely successful aside from sometimes being able to appeal to a very niche crowd for the somewhat abstract nature of the entire affair. Most of us though, appreciate the meat on the bones that is formed by all those extra details. To draw this analogy to its logical conclusion however, too much meat makes for a bloated, fatty and sometimes unduly poor experience. It's striving for a balance between the elements, building the Adonis as it were, that games should aim for. Going from this point of course there will be instances where some things will obviously work better for some genres than they will for others.
For example: I don't think anyone really minds a story in an RPG; in some cases that's the actual draw as much as gameplay graphics or any other element. An RPG which has a grand story that is told with epic grandiosity can make a player feel like they are truly part of a legend, if only for a fleeting moment. But a grand narrative will be tolerated and generally accepted more from say, one of the Final Fantasy series than it will be for a game like Banjo Kazooie. For action games, or adventure games or most others a story is needed, but can be dialed back to a basic premise "something somewhere is wrong, get out there and fix it because that's what you do". Likewise, in some genres story can just seem downright stupid: puzzle games often don't need a story; do you really need a motivation to play Tetris beyond the fact it's Tetris? (if you answered yes you may need help) In fact, giving a story to something that really doesn't need it can do more harm than good, that is if the audience doesn't just outright ignore it entirely.
I think one of the main problems these days though, is that in some cases story in games almost seems ... masturbatory. Sometimes game designers have stopped seeing story as one tool among a plethora of ones that are used to make a game better and instead see it as something that a game has to have or worse, as something that they include because they get enjoyment out of it regardless of whether the person playing the game will feel likewise or not. Story is not something to be foisted upon the audience at the expensive of overall experience. Some games have had pretty nonsensical stories and have suffered for it: I'm looking at you Sonic series, and to a lesser extend even games like the recent entries in Assassin's Creed which have gotten kind of off-the-wall out there in terms of narrative. In the latter case the core gameplay experience is still solid enough to forgive these sins, but the story will definitely not be remembered as the high point of the AC series anytime soon.
What I'm ultimately getting at is that story might feel like something that is absolutely necessary, since games these days generally have higher production values than at any previous point in the industry. There seems to be a natural urge to attempt to give a game a blockbuster style story, or something that competes with movies or books or television. In attempting to get to this level though, we might end up losing sight of something important: we play games. We interact with video games more than nearly any other type of medium. We are passive when we watch a movie, read a book, or tune in to our favourite show. When we play a game though, we are the ones driving the action: things only move forward as we do, and it would be wise to keep that in mind, and balance that thought against the seemingly ever increasing desire to turn games into elaborate stories at the cost of what endeared us to them in the first place.