Although not exactly a “game” in the sense of other games listed here, I just had to mention this product, the Aerobie brand Dogobie disc. We have a large Aussie, and he tears up regular throwing discs in minutes. This disc has lasted for years.
Our 80lb Aussie dog loves this disc, and it is safe as well as durable. Most throwing discs are hard plastic, and can cut the gums of any dog that bites through the plastic. Or, break a tooth when a dog catches a fast throw. I personally much prefer this softer disc, to minimize chances of injury to our dog.
Granted, this is not a disc for long-distance throws or extreme accuracy. It throws well compared to most consumer discs, but the range is fairly short. It is made specifically as a dog throw & catch toy, so don’t try using this as a driver at a disc golf course!
Although our Dogobie disc does show hundreds of little indentations from tooth marks, Ranger has never bitten through it. The raised rib on top makes it easy to pick up, plus it floats, and comes in 2 colors that even color-blind dogs can see.
While these Dogobie discs are very durable for toss & catch play, keep in mind that these are not intended as a chew toy. If you just leave it with a dog to gnaw on, it will be ruined just as fast as any other dog toy.