Bear with me here, there is a point buried in all the squiggles.
I think.
One of the characters I play as is an Imperial named Ulysses (named after the latinized version of Odysseus, NOT the dude from FO:NV). Like his namesake, he's a trickster type: he avoids any non-essential conflicts through judicious use of Illusion, the Throw Voice Shout (especially in combination with traps

), stealth, poisons and general cunning. I think I've killed about 10 enemies per level so far (I'm level 40 at the mo') and its been a heck of a lot of fun.
As a trickster I've found a neat personal justification for his eternal servitude to all the Daedric Princes by looking at another very cool trickster from the world of comics: http://en.wikipedia.org/wiki/John_Constantine.
Nononono, not bleedin' Keanu. Look, here he is:
http://www.scifinow.co.uk/wp-content/uploads/2012/11/Hellblazer-cancelled.jpg
Much cooler than Keanu, right...?
SPOILER ALERT FOR HELLBLAZER:
You see, during Garth Ennis' run, the chain-smoking, smart-mouthed, sweary, mage John Constantine contracts terminal lung cancer. Bugger. This comes after he recently pulled the wool over the eyes of one of the three Lords of Hell (known as The First), who vowed to claim John's soul upon his death and use it as toilet paper for eternity. To clarify: the Lords of Hell are essentially in charge of all evil and therefore are definitely not to be messed with...
...unless you're Constantine.
Needless to say, John doesn't really want to have his eternal soul fall into the hands of a very angry Lord of Hell (or any other appendage/orifice, for that matter), so he sells his soul to the other two Lords of Hell on the sly. When the First Lord comes to claim it, so too do the others.
John's soul (like I imagine the Dragonborn's to be) is so important that the Lords are literally obliged to go to war to keep it. Such a battle would devastate Hell and allow the forces of Heaven to finally come down and destroy all evil. The Lords of Hell, after much vicious debating, decide to cure John's cancer (in the most painful way possible) and keep him alive, thus avoiding any unwinnable conflict.
As he leaves, John turns to the three Lords and... well, take a look... go on, I'll wait:
http://i436.photobucket.com/albums/qq84/DurrHurr/characters/upyours.gif
So, bringing this all back to Skyrim: My boy Ulysses has sold his soul to every Daedric Prince and tricked them, gaining their gifts without having to ultimately pay for them. If Ulysses dies, those Princes are going to have a "Hell" of a time fighting over who gets my trickster Dovahkiin's ever-lasting soul.
TL:DR - Dragonborn sells his soul to all the Daedric Princes, so that none may claim it without having to go to war with the others. Or something.
At least, that's how I justify it.