I tend to be more of an action shooter and our private range has lot's of steel so most of our games center around normal steel stages. Just like real steel matches we use a times and fastest with no misses wins. My personal favorite is El Presidente.
We do some target work with our .22s and potatoes, Old swing set with strings hanging, drywall screw tied to the strings and slide on a potato. We fire from 15 yards, each person taking turns with a single shot. whoever gets the most out of 10 wins.
We also do a rifle game (rimfires again), found some targets with 8 bulls on each. Set them up at 50yards, call start and first one to hit all 8 bulls wins.
One thing is many ranges will not allow many of these games. That is why I consider myself very lucky to have a private range were pretty much anything goes.