This post is very subjective.
If you like to hunt wild hogs, one of the southeastern states or Texas would be the best.
If you like black bear, Maine is one of the best states to live in for that. For deerhunting, the further north you go the chances lessen due to the winters. If you are lucky to get a permit, moose in Maine, New Hampshire or Vermont cannot be beat. But my friends wife finally got one after applying for fifteen years. So, don't hold your breath waiting for a permit.
Deer hunting. Again. That may include the midwestern states like Ohio, Indiana as well as Illinois. The prairie provinces of Canada have produced some real monsters. The South has the greatest densities so your chances may be much greater.
Elk. I'm kind of clueless here. I've never hunted elk but I remember seeing a TV show on elk hunting in Kentucky. If you're lucky to get a permit, your chances are nearly 100%.
Another thing. If you're a handgun hunter. The US is it. Mexico and Canada are off limits to sixgunners and TC addicts.
Again, this post is very subjective.