Looking at the suggestions so far, I'd say Scott's two suggestions take the cake for oddity. You certainly won't run into another hunter carrying a double-barrelled or three-barrelled rifle here in the US.
Myself, I like my guns to have a little historical significance. So in that department I'd say the Winchester 1895 and the M1917 Enfield come in tops.
Big and ugly, the 1895 was one of Theodore Roosevelt's favorite guns. Although not currently in production, FN-USA does produce limited numbers of this rifle from time to time. A used 1895 in decent condition will set you back a grand or more.
If the 1895 is a little too rich for your blood, then consider the M1917 Enfield. Noted for both accuracy and ugliness, it was the Enfield, not the Springfield, that most of the doughboys carried in France. Unfortunately, many of these fine rifles have been sporterized, which does little to blunt their homeliness.