First, we need a bunch of assumptions.
The resulting super-stretched Falcon upper stage / LEO-out booster should hold 475,000 kg of propellant, making it about twice the size of the Falcon 9 lower stage. If built with the same tank diameter (3.6m), it would be about 56m long. At launch, the whole thing would be about 89m long, with an aspect ratio of 25, which is pretty skinny but probably doable.
Such a booster could give 4,000 m/s delta-V to a vehicle with a empty mass of 168,000 kg. That's enough delta-V to get out of Earth orbit to L5, low moon orbit, and at least near Mars, see this handy cheat sheet. It's also just a bit more mass than the Saturn V rocket put into Low Earth Orbit -- not excessive for a manned excursion to anywhere dramatic.
It would require 49 fuelling flights, with a total cost (just for the booster) of $1.35 billion dollars with SpaceX's published prices. SpaceX's prices are for single launches, assuming no hardware recovery/reuse. 50 total launches would get them a lot of experience with recovery, and change the whole cost dynamic. It would also take at least of year of fairly amazing launch activity.
An equivalent booster, launched with the Shuttle-Derived Heavy lifter, would require about 3 launches, as it uses liquid hydrogen, which really helps this application. Shuttle launches are hard to account for, but probably around $500M each, and if the SDH is similar, it would cost $1.5B for this booster, not counting amortized development costs.
I don't think anyone would start an Artic expedition that required 50 plane trips just to build a cache of dog food, so I don't think this version of the idea is going to fly. Perhaps liquid hydrogen is inevitable for this application. Alternatively, SpaceX's Merlin 2-based launcher may get launch prices down enough that Kerosene and LOX make sense even for a big out-of-LEO booster.
[Note: the next post answers comments.]