The Football Manager series (formerly known as World Soccer Manager in North America) of managerial simulators gives you a fair amount of leeway to abuse any player on your team. You can bench someone, stick them in the reserves, yell at him, smear him in the press, work him to death during practice, retrain him as a goalkeeper, fine him two weeks pay over and over again for any reason or no reason whenever you feel like it, and generally do everything in your power to make his life hell. Like real life, these tactics are often used as a deliberate strategy to get a rid of a player they don't want on their team any longer, by trying to make him so unhappy that he'll finally agree to a transfer or having his contract bought out.
The NFL Blitz series of football games allows many types of hits and tackles that would have players fined and suspended, at minimum, if they did them in real life. It's certainly not necessary to take the ball carrier down with a flying clothesline and then suplex him two or three times after the whistle, but if you feel like doing it, hey, have at it!
After play stops, the defending team can physically abuse opponents until the game cuts to the play screen or fades out (at the end of a quarter). End of the game and the team on defense just suffered defeat by Curb-Stomp Battle? Time to take it out on the ball carrier!
Blitz: The League, which is pretty much a NFL-less Spiritual Successor to Blitz after EA secured exclusivity of the NFL license, is this trope in spades. The fact that you can do brutal injuries that would be considered career ending to other players and be rewarded for it just drives the point further, as well as sabotaging the other team by sending them prostitutes and giving your team drugs that may be illegal. Blitz: The League 2 takes everything about Blitz: The League and turns it Up to Eleven.
The Slugfest series, also by Midway, basically did for baseball what Blitz did for football. The pitcher even had an automatic bean button. It could never be considered useful in normal play, but boy is it funny.
The Skate series. The first one had Hall of Meat, which told you how many bones you had broken. But things get amped up in Skate 2, where you have the ability to trigger bails at anytime. And then there's the insane physics engine in Skate 3.
Extreme Warfare Revenge, a Professional Wrestling managerial-simulator, also has ways to abuse the wrestlers that work for you. If you just don't like them, give them an insulting gimmick and make them lose to everyone. If you've got a wrestler on an expensive guaranteed contract that you want to get out of for some reason, just promote him up and down the card until his morale hits bottom, and then he'll quit so you don't even have to pay him his severance fee.
In the 1990s skiing game MacSki, you can crash the skier into as many obstacles as you wish. It'll take him some time to get back up again, and a St. Bernard will rescue him after every fifth crash, but there's still no limit to how much you can do to him.
Some courses have other skiers, snowmen, bunnies, and penguins for you to run into. The other skiers will let out a "Hey!" when you run into them, while the ski bunnies will shout one of five random insults such as "Bean brain!" at you.
Some go even further by making it a course requirement that you crash into the snowmen or penguins. That's right, a skiing game that lets you slaughter penguins.
The Tony Hawk's Pro Skater series started adding NPC pedestrians from the third game and out, and nothing is stopping you from ramming (most of them) down while skating around. Some missions even require you to do this, such as in the fourth game where you have to knock a whole bunch of tourists into shark infested water just because Bam Margera told you to. Underground 2 added throwable weapons and American Wasteland added in a feature that allows you to use your skateboard as a melee weapon when on foot, and both features seems to have no other use than to attack random NPCs for fun outside of a few special missions. Tony Hawk's Project 8 even rewards you with tokens needed to buy special tricks by knocking down other skaters, even showing them raging at you after the attack in a popup camera window.
In Arc Style: Baseball!! 3D, when playing as a pitcher, you can hit batters in the face on purpose, and no punishment will occur. Besides walking the batter, that is.