Bighorn sheep...but unless you have an unlimited wallet then it's maybe a once in a lifetime thing... I'll break these out by budget...my first list is sub $10k...something if someone saves can do this type of hunt every year or at least every 2-3 years....
1. Aoudad sheep...$2500 to $5,000...awesome hunt just like a desert sheep hunt...I have taken countless Aoudad-I was born out in west Texas and own and access a lot of good Aoudad land...would do it everyday if I dint have to make a living...love it!!! ( semi guided...to guided..lots of good guide options..will have a great 3-4 days out in God's country)
2. Nilgai Antelope in South Texas--fun social hunt...good to do with customers or close family members--1-2 day hunt, Guided is the way to go..Double-shot outfitters, Pierce trophy hunts--are good options down here. Cost $2,500-$3500.
3. Pronghorn Antelope--done it lots of place and enjoy it...spend a lot of time behind the glass..peaceful...unless you are there opening day...then it's a rodeo on public ground. $500-$2500
4. African Plains Game safari to Namibia or South Africa...best bang for the buck in the hunting world...hunt for 6 days kill 5 animals--eat like a king for $5k-$7k--it's a true vacation--lots of good options Kowas safaris, Spiralhorn Safaris are two really good options.
5. Good ol whitetail deer...what's not to like!!! Free-to--$$$$$$$
Above $10k
1. Bighorn Sheep/Dall/Stone...love it done it twice and would do it every year I could. Lots of good outfits if you can draw a tag...some of Dall/Stone Prophet Muskwa is a good outfit...cost $15k-$100k
2. International sheep--Marco Polo, Gobi Aragli, ect....same as above...
Do your research and enjoy a quality sheep hunt. $25k-$200k.
3. Cape buffalo/elephant. Dangerous, adrenaline filled fun..lots of good places...done it twice...price $12,000 to $50,000 depending on where and length of time.
These are my favorite. I have hunted cats (leopard)..and it was an experience but not my cup of tea. Hunted elk and moose--yes it's grand sport--but a lot of work begins when the critter is down--bear was fun once--but did not light my fire. Lots of things I still would like to hunt...the Capra species is really interesting to me--Mid Asian ibex, and Marco Polo is high on my list, . So is the ibex slam in Spain although they seem a bit pricy on these hunts. I'd like to draw a New Mexico Ibex tag and do that hunt there--but that tag is about as hard to draw as a bighorn!!! To much to do...so little time on this earth!!!