A tough question, hope to be able to chase goats seriously some day if I ever draw... I have been plenty close to them in MT and CO on occasion minus the license, bow and snow...
Cougar w/out dogs, tried, got close but no cigar, maybe some day, hopefully before I get too old and fat to run one down...
Southern AZ Coues, been once, saw several, huge fun, didn't bring one home but sure plan to go back for another try on those wary critters one day.
For sure the toughest critter I've personally chased has to be the "educated" wolf. Prior to MT allowing hunting of them they were pretty easy to call and get reasonably close too. Now after a few years of education they have learned well. Unpredictable, naturally wary and the ability to travel unbelievable distances in a short time make them one tremendous challenge for the stickbow shooter. Spent countless behind the binos and spotting scope looking and planning the stalk. Been close several times, even within touching distance once but only had two shots. One a complete miss the other a nine yard gimme, he just wasn't there when the arrow got there. Truly frustrating but true fun at the same time. Can't wait till next fall!!