Mine would be:
.223 in a nice AR built for defense, but they work great for coyotes, PD's, and fun shooting
LSR 7WSM for game from pronghorn to elk
338 Edge for anything from deer to bears, heck I'd have to try it on a few coyotes too!
The 223 could be replaced with a precision 243 for a more rounded game trio, but if I were limited to 3, an AR would be vital if the poop ever hits the fan.