In the 1950's an indian woman in Canada killed a large griz with a bolt action .22, dropped it with one shot to the dome at close range. Many people contend that every large game animal on the planet, likely including elephants, has been killed with the humble .22 .
That being said, I wouldn't want to go into grizzly territory with even a .357 .
I hiked out to the "magic Bus" last year, it's amazing that he was less than a days hike from civilization and still died there. I carried my .44 Mag, I wasn't going to test the theory of the .22 if it came to that.