My experience with quests, and indeed most people's experience with quests in MMOs to date, was pretty bland for the most part. Sure, some incremental progress has been made beyond "kill ten rats", though those quests are still around, since they make killing ten rats feel slightly more productive. There's the occasional quest that's actually enjoyable; sometimes it's the unique mechanics, like driving around in a siege engine; sometimes it's the story, like the Wrathgate questline in WoW.
However, quests are all game stories, without exception I believe. They were all stories that are built into the game, were written by someone else, who tried to make them in such a way that you wanted to know them.
I have told stories about playing WoW many times (though usually to other people who play WoW), and I'm sure most people have a few good stories from playing an MMO. However, I'm willing to bet that none of those stories are a quest story. Oh, they might be a story that happened while doing a quest, but they won't be the quest itself. Why retell something that's been told to literally thousands of other players in exactly the same way? My favorite stories are all of times when I did something cool with other players. The time my guildies and I ran Stratholm deadside in 45 minutes in blue gear. The time myself and 14 other PvP-focused players running a premade defeated the strongest guild on our server in a hotly contested AB battle that ended 2000-1990 (vanilla WoW; the guild was in T2.5 and we were in PvP blues and epics).
You know what I conclude from that? The best quests are the ones that make you interact with other players, then get out of the way. I have virtually no memory of the story behind the quest to run Strath deadside in 45 minutes, but I remember many details of the run itself. I may not remember many of the details of the 2000-1990 AB fight, but I do remember the desperation and then excitement upon realizing we might be able to come back from a deficit and win.
I'd like to propose an example quest that could do this. Imagine a quest in town to escort an NPC to the next town, perhaps a 15-20 minute journey. The quest is only offered once, and when someone accepts, no one else is allowed to sign up, similar to some escort quests in WoW. However, imagine it's an open quest; you can optionally share it with as many people as you like. However, sharing it may reduce the reward slightly, the idea that the payment would be divided among several players.
Imagine, next, a corresponding quest in the nearby enemy faction hideout. At the same time that the escort quest is accepted in the nearby town, the hideout has an NPC start offering this quest to any single person interested to assassinate the NPC who is being escorted. This player can choose to share the quest with others, with the same sort of minor penalty. Alternatively, maybe a standard Wanted poster is up with a reward for any player who turns in his head.
The point here is to create a situation where players have a reason to attack each other beyond simple "ganking". With a structural reason in place, preparation can come into play. A player escorting an NPC with a bounty on it's head knows that there's a high chance they'll be attacked. This creates a more equal ambush situation, because the player is expecting it, though he still might be taken by surprise. Also, there is the interesting decision of whether or not to bring along help. The escort player may opt to not bring anyone, on account of the zone being under his faction's control. However, he knows that an enemy in his own zone might be able to kill the NPC before he can stop them. Similarly, the enemy player has to decide whether to bring along friends and increase the risk of detection by random players, or go it alone and risk simple failure, or risk being outnumbered by the escort.
I'm sure there are other many ways to vary this sort of a quest, including ways to make it more attractive to pursue. However, I think the general theme of structuring quests in a way designed to throw players together and at each other needs to be put to more extensive use.