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You specified DANGEROUS game & wanted our thoughts on a minimum chambering. If you go to book a Alaskan Brown Bear hunt most outfitters will specify a minimum of a .338 Win mag. I know this because I was just shopping for such a hunt. Several said they wouldn't take any client carrying less.
For me, the minimum for dangerous game would be .300 RUM or .375 H&H. Both these exceed the power of the .338 Win.
The .300 RUM is not less, but it is different. It carries more energy than the .338 WIN, along with a flatter trajectory in case a longer shot presents itself. The .300 RUM is no fun off the bench, but I couldn't imagine anyone noticing the recoil with a big bear in front of them.
The .375 H&H is chambered by every major gun manufacturer and ammunition can be bought anywhere from Peru to Finland. It's recoil is described as more of a shove than the jolt from a .338 RUM or .340 Weatherby, very similar to a magnum shotgun.
If longer shots were the expectation, my minimum would be the .300 RUM. If shorter to mid-range shots were the expectation, I like the added bore of the .375.
With any magnum rifle you will need to practice. Just don't start out trying to fire 2-3 boxes of ammunition at one setting. If you do you will likely teach yourself to flinch.
Good luck & have fun!
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