The hunt that I believe would be most impressive to do with a handgun is definitely a mountain sheep/goat/ibex/etc. with a revolver. Climbing plus stalking plus weather plus altitude means you need to be a hard man (or woman) to do it.

For me, the dream has always been Africa. Of course, if I could take my modern handguns but hunt in the traditional early 20th century style (long hunts, big camps), that would be the dream.

Coming in at a close second would be a free-range bison hunt.