I think you have multiple questions pretending to be one:
1. What is the most physically demanding game animal in N.A. to hunt. I would say anything where there is not enough oxygen for me to breath...goats, sheep, etc.
2. What is the hardest tag to draw in N.A. Of course, since every state is different, that one is going to be confusing. In Oregon, I feel like Pronghorns are a lower chance of getting a tag than winning powerball since I have been trying literally my whole hunting life and have not got one. Of course, I could travel to Wyoming and buy one over the counter.