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Montana offers attractive draw hunts for premier species of Rocky Mountain Bighorn Sheep, Mountain Goats, and Shiras Moose. The draw odds are tough, but at "only" $750 per license, these tags are relatively attractive (the $750 per tag is refunded if you fail to draw). Stick with it for a few years to save up bonus points (participating in the bonus point system for the premier hunts costs a flat $20 per year), and it may be possible to draw one of these tags in Big Sky country.
Montana hunts for deer and elk are, for nonresidents, more complicated and more expensive to apply for, and are certainly less attractive than those offered by other states.
Montana has some good antelope hunting, but Wyoming offers such good antelope hunting that I've never felt the need to travel further from home (for me) to hunt in Montana.
Starting in 2005, Montana also began offering a late season bison hunt. One of the few free-ranging bison hunts in the Lower 48, this is again an attractive draw hunt at "only" $750.