I typically shoot 3-shot groups for testing hunting loads. I do not do any target or benchrest competition. Three shots gives a pretty good idea of practical hunting accuracy, and three shots generally gives you a good idea of whether or not you need to change your scope settings. If you can get your groups down to MOA for 3 shots then you can have some confidence in them as far as repeatability and scope settings.
For a lot of shooters, myself included, I believe that it's hard enough to really concentrate on sight picture, scope/parallax control, breath control, trigger, consistently holding the gun on the bench, etc., for 3 shots and with 5 I'll almost certainly let one off when I wasn't ready. If I feel that I threw one shot out of the group, then I may fire one more.
If you don't think that the number of shots in a group matters, get an accurate .22 rimfire - a true "one holer", and work on shooting long strings (say 10 shots) and that will tell you a world about your ability as a shooter to really concentrate for 3 shots vs. 5 shots or more. Hey, if you can do it with long shot strings, great!
Plus - it does save bullets. The last several times I have worked up handloads, one time it took 10 bullets to get an acceptable handload and scope setting for a .458 Win Mag - and you don't shoot that gun from the bench more than you have to! And another time it was 12 bullets. Of course some of this is luck, but if I have to shoot 10 groups, then that's 30 bullets vs. 50 bullets and I have 20 bullets left in the box to shoot pigs, deer, etc.
Of course you occasionally have to chase bedding or scope problems, I ran probably 120 or more rounds through my .35 Rem till I got it sorted out. Fortunately, factory ammo is cheap for that round.
It does depend on your gun - a .35 Rem at 33,000CUP doesn't heat up a barrel like a .22-250 at 50,000CUP, as you might expect.
As I understand it, when the steel gets heated to a certain point, it loses carbon and/or temper, then softens and gets washed away by the bullets/powder gasses.
You can think of it as spot-annealing, in a microscopic scale, if that makes sense. A little spot on the lands of a rifle or in the throat is going to absorb a LOT of heat real quickly, then dissapate it to the surrounding steel. But dissapating heat takes time, it isn't instantaneous.
I'm sure the manufacturers don't publish any sort of information on this, there are too many differences in individual barrels. They all aren't going to be the same hardness, they certainly won't all have the same surface finish (and friction with the bullet does contribute a lot of heat), then you get into differences in powder types (ball vs. stick, single-base vs. double-base), and how dilligent each person is about keeping the barrel clean. Same reason most manufacturers don't publish accuracy guarantees, either.
Don't let the thought of washing a barrel out discourage you from practicing, though - use either lighter loads (reduced loads with cast bullets and barrels last a long time) or break out the .22.