Of all the animals that I've actually hunted successfully, (mule deer, whitetail deer, antelope, black bear, elk and feral hogs) I would say that Elk are by far my favorite. They truly are majestic creatures. Plus they taste great and they pack around plenty of meat on their bones!
I have an itch to hunt mountain goats right now though, folks who I've talked to that have hunted them say they are one of the most enjoyable animals to hunt. I don't know yet, but would like to find out sometime soon. I've got some applications out there right now, maybe in a few years I'll get lucky.