Hope I'm not being hoggish, but you did ask for three, and it could be done on one trip whether I was successful or not. Just the trip and the experience would be enough.
1. Alaska, Upper B.C., Yukon, or NW Territories on a guided hunt for moose, grizzly, and sheep.
2. Bighorn ram here in Montana DIY
3. Grizzly bear hunt here in Montana DIY
To that end I have to add that my mountain goat hunt last fall was incredible even though I never got a goat! I loved it and the time I spent, most of it by myself, in the mountains!
David