Another possibility is that there is _not_ enough dark matter to halt expansion of the universe. Here we would probably be saved by the n'th law of Thermodynamics (n'th is used to indicate that I can't remember exactly which one it is I mean here, so you have to figure it out yourself). Basically, this law states that all closed systems tend to entropy, which is a long-winded way of telling us (so I'm told by all those really depressing stories I keep reading in Discover magazine) that the universe will eventually all end up spread out as a cold and lifeless collection of widely spread out nothingness.
Given this scenario, at some point there wouldn't be enough energy left to power your machine, so there wouldn't be any point in you actually storing any more URNs anyways. Again, we don't know how long this would be yet, but since we started with a finite amount of matter and it's expanding (and cooling) at a finite rate, we've a finite amount of time before you might as well unplug the machine and go home.
Assuming we can calculate how long it takes to get everything to cool down enough we can thus still put an upper bound on the number of digits we need to allow for (and I understand that now the superconducting supercollider's been cancelled there are thousands of out-of-work physicists who'd love to help with the calculations on this in exchange for having their name on the resulting paper. As the supercollider's budget showed, these guys really love big numbers but I still think we're talking a finite number of digits here).
The only possible wrinkle we can't rule out is if there is _exactly_ enough dark matter out there to halt expansion of the universe at some still useful point on the thermodynamic power curve, but _not_ enough to cause everything to collapse back down onto itself. I know this sounds unlikely, but who'd have predicted the success of MS-DOS ten years ago? Given this existance proof of a capricious and vengeful God, we can't rule this third possibility out.
Still, covering two out of the three possible bases in a version 1.0 URN still seems better than any of the other proposals we've got kicking around right now. Besides, I think we'll be proposing something that would last a bit longer than the address space used in the initial version of IP. And if turns out that MS-DOS wasn't a one-off mistake, we can plan on fixing our problem using the same technique they come up with for IPng.