Bear in mind that double rifles became dominant in the 18th and 19th centuries, when reliable repeaters were simply unavailable. (In blackpowder days, of course, the doubles were muzzle-loaders anyway! It's a big thing to have a second shot immediately available, rather than waiting a minute or so for the reloading process to be completed!
) When the first repeaters came along, their actions weren't strong enough to handle big-game rounds. Only with the advent of the Mauser action did a repeater become available that could handle the really large cartridges.
As a result, most of the big-game cartridges that were developed were deliberately optimised for the doubles: large rims, cartridge design not necessarily suitable for magazine loading, etc. If you look at a .470 Nitro Express round, and compare it to the ballistically-similar .458 Win Mag, you'll see what I mean.
The doubles offered several advantages, which others have mentioned above. Their speed of operation was also legendary: I've watched professional hunters fire two, reload, and fire two more, accurately, at 50 yards, and beat competitors firing a bolt-action repeater next to them. Being from Africa, I learned the old technique of holding two spare rounds between the fingers of the support hand. You fire two, break the action (which, of course, has ejectors rather than extractors), dump in the two spare rounds, and close and fire the gun. With practice, it's very fast indeed. (Also, the bell-like TONK! of the big fat rounds dropping into the chambers is music to the ears...
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However, as noted above, the double is a VERY expensive rifle: not only were most of them hand-made rather than production-line guns, but regulating the barrels, etc. required immense attention to detail on each gun. A quality double will put a round from both barrels within 2" of each other at 100 yards: some will do it in 1". To get this degree of accuracy is tough!