I think in general the story wins out. Often campaigns end before the character in question dies and the goal he is working to may be unresolved. For example if I have a backstory that says I am a changeling at birth and will some day go back to the feywild to confront my parents, does it matter if that is resolved in a campaign that mostly focuses on goblins?
In the mind of the player he could always write his own epilogue after the completion of the campaign.
This is the sort of player pragmatism that I so rarely see.
So many things can go wrong with player trumping the DM, not with the DMs fun, but with the player's fun.
a) It presumes that the story the player wants is actually one he's going to enjoy more than the one the DM wants.
b) It presumes the player is actually a decent story teller.
c) It presumes that the other players are actually going to enjoy the player's story more than they'd enjoy the DMs story.
d) It presumes that the player's story is compatible with all the other player's stories. At an extreme, it presumes that the story the player wants isn't, "I'm secretly a traitor to the party.", "I'm going to kill all the other PC's".
e) It presumes that there is actually space in the campaign for telling a story focused individually on six different PC's goals.
f) Individual player goals almost always splits the party, imposing difficulties with running the story as forcing players to watch other players play.
g) It presumes that the player actually has a functional model of play as opposed to wanting to DM from the player chair, wanting to always get his own way, wanting to always win, wanting to always win easily, wanting to always have the spot light, wanting to always one up the other players, etc. I've seen literally seen players combine the personality of, "I'm always wanting to win", with the backstory of, "I'm secretly a monster who is going to kill all the other PCs". Yeah, that works so well for everyone...
h) It presumes that the other players aren't expecting the DM to deliver a story, and will be perfectly happy to have another player in the driver seat trumping the DM's story.
I) It presumes this particular PC is going to survive long enough to not only have a meaningful impact on the game, but develop a satisfying story line.
j) It presumes that the campaign is going to go long enough that not only can this one guy fit his story arc into it, but everyone can.
In my experience, 4 times in 5 the guy running the game is the best RPer, most experienced player, puts the most effort into the game, put the most thought into everyone else having fun, and has the most mature outlook regarding table conflict of anyone at the table. It's rare that I'm sitting at a GMs table and I want him to give in to another players demands and ideas.
Right now I have a glut of potential campaigns I could be running based on player backstories. It's taken 4 in real life years just to get where we are now. There is no way I'm going to resolve every possible story line based off of every hook the players have thrown me, and that's to say nothing of effort that was expended on integrating character backstories in to the game only to see the character die off before anything could really come of it leaving orphan story lines where no existing character knows IC about the NPC's and conflict involved. And right up on the horizon I've got a major worry as a DM where one character's backstory is going to force the PC to make a choice between pursuing the villain the players have been chasing for 4 IRL years, or letting that trail go cold and pursuing something of overwhelming personal importance to the PC. It could make for great dramatic RP. It could also completely suck for not just that player, but the whole party.