Alaska - you can hunt brown bear/grizzly and sheep with an over the counter tag in many places.
If you go down to Southeast Alaska, black bear are as common as racoons in Ohio.
Brown Bear - Kodiak/AK peninsula
Grizzly - Brooks Range
Dall Sheep - Brooks range
Deer hunting - Kodiak.
Caribou - Mulchatna
Moose - Western Brooks Range
Elk - Kodiak again
Goats - Southeast or Kodiak
Coyotes - Kenai
Wolves - anywhere
Did I forget anything?
The best hunting in Alaska is expensive though. There are no roads to speak of and you'll find that if you can drive there, you will find competition. Spend the money for a fly-in and you can get hunting that rivals anything on earth.
The best combination fly-in hunt is probably Southwest Alaska. Without a drawing, you can hunt Moose, Caribou, Grizzly, Black Bears, Wolves in the same area. Pretty hard to go wrong!
Alaska probably also has some of the best duck hunting in North America as well as ptarmigan, grouse, rabbits, etc.
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Keith
The Bears and Bear Maulings Page:
members.xoom.com/keithrogan