I know several great ones. Some definitely specialize, and are exceptional at specific animals, while others excel at others. Anything in particular you are looking to have done?
Also, keep in mind, a bigger name/highly reputable guy might cost a little more. But this site is littered with posts of guys who had stuff ruined, money lost, etc. by small guys who closed up shop and vanished, or ruined stuff.
My votes for the top all around guys would go to:
Clay Goldman at Mogollon Taxidermy, (Payson, AZ), Fantastic all around taxidermy. IMO, the best deer and elk guy in the state. Only person I am aware of in the state who has the machine to do velvet antlers in house (legit machinery, not injecting of sticking in a freezer for a LONG time). He is currently doing my bull from this past fall.
Dale Robertson at Wild Heritage Taxidermy, (Young, AZ), I don't think he is taking new clients at this time. Fantastic all around taxidermy, but he is the best lion and cat guy in the business. I haven't seen ANYONE in the country who can consistently do the quality cat work Dale does.
Richard Owens at Hunter's Lair Taxidermy, (Show Low, AZ), Great all around taxidermist and is currently doing my son's bull you see in my avatar picture.
For birds, I recommend:
Kyle Thorn at Cut-N-Strut Taxidermy, (Surprise, AZ), He used to do all birds, but is trying to focus solely on turkeys these days. He competes with the birds he works on and wins a lot of awards.
Sven Eberson Wildlife Wings and Design Taxidermy, (Glendale, AZ).
Skull cleaning and Euros:
Draysen Hatch at Great Hunter Skullz, (Queen Creek, AZ), I am highly biased here (he is my son and I taught him), but I don't think anyone does a cleaner and whiter skull, and I have seen a lot.
There are, of course, many other great ones in the state. I am by no means knocking any others. But I have seen first hand and done business with all of these guys. I have always been extremely impressed with everything I have seen done by them, and could provide personal photos from my family's collection from each of the above mentioned.