I understand everyone's points and they are all valid (I LOVE the feathers in the vase trophy by Muley, that is an AWESOME idea and can be a family heirloom for generations!) but IMHO the difference between trophies with regards too elk vs deer is two entirely different things. A 180 class buck is MUCH more rare than a 300class bull. I think something more befitting would be a 380" bull than a 300"er. Another thing I personally feel is that an elk on "public land" is no different than a bull on private land in most locales in the west. The simple fact is that most elk (and even deer for that matter) are not going to spend their entire lives on either public or private land so I think that fact alone negates the "public vs private land" arguement for what makes a trophy.
Maybe in some heavily hunted areas like many in CO your chances are substantially better on private than public but in better managed areas like AZ, NM, UT and a few other places, the bulls are just as likely to be on public land as anywhere else. Afterall, there aren't that many 100k+ acre ranches out west. Landvalue and inheritance taxes have seen too the selloffs of the largest ranches that were around 75-100yrs ago. I suppose the largest "private areas" are the various Indian reservations with the best obviously being those in the San Carlos/White Mtn. Apache reservations in AZ. But do I consider a 320"er from CO a better trophy than a 400"er from White Mtn.? Certainly not, a 400"er is the thing dreams are made of and what ALL elk hunters desire whether they want to admit it or not. You can simply answer that question by asking one simple question, "You are in XYZ state and have a tag in your pocket and suddenly you have a 400" 7x7 and a 320" 6x6 both standing in the same pasture and both easily within gun reach. Which one you gonna take?
Unfortunately we can't all hunt White Mtn. so I believe that that leaves alot of "CO DIYers" too mask their fate and call the hunters who CAN hunt the prime areas, somehow less hunters or dismiss their trophies as being "less" than some raghorn killed on Federal land in a shotout region of a shotout state. In reality that has NOTHING to do with the "trophiness" of either hunters animals.
As for the deer question, a 180" buck (either muley OR whitetails) is one tough animal to come by and you will spend years and ALOT of money looking at many many deer before you might even catch a glimpse (if ever) of a unique monarch. A 180" buck is a 180" buck IMHO and it is so just about anywhere it lives. I suppose some small "pen" of say 1000 acres or less that is behind a fence COULD be considered cheating. But anyother size plot simply is ALOT of ground for such a buck to spend his life. The big ranches in southern TX are simply AMAZING plots of ground and for those that haven't had the luck to hunt on one or two of them, they are not the "answer too all questions" many would have you believe. A 180" buck for most intents and purposes will be a mature animal that quickly learned the score and is no dummy.
My definition of a trophy actually depends on the animal itself. For whitetails I consider an 8ptr over 140" to be a trophy, anything over 10pts needs to be 150"+ before I turn the safety off. But that obviously comes after 2 decades of chasing them and having the luck to have taken several animals of nice size. A 160"+ ANYTHING is going for a ride in my truck if I see it!
For bulls, I have a wonderful "Gila monster" on my wall that is a 368" 6x6 (the Gila/Apache Nat Forrest is public land btw) when I return too that same area (hopefully by bow, my other was by smokepole) any 6x6 over 320 is gonna get perforated. I have a unique desire for a MONSTER 5x5 and would love to have one in the 340" range. Any 5x5 over 300" would be a rare and unique trophy in my opinion. As for bulls any larger... get out the skinnin knife!!!
If a 4x4 raghorn taken just west of Denver is your definition of a trophy, then who is anyone else to say otherwise?

RA
ps
As crazy as it may sound, I can assure you that Chuck Adams would NOT shoot a 340" bull anymore. He is in the position that assures that he aint "chasin bugles" in overhunted areas anymore, he is hunting the primest of the prime areas in the country. I doubt seriously he even draws on anything under 380"! Nice "problem" to have huh?