I have only hunted with single shots. On only one occasion have I felt that I was at a disadvantage as a result, and decided that for close-quarters encounters with dangerous game, I would opt for a repeater (and a sidearm, and a tall tree, and a box of baby-wipes).
While you can flawlessly execute a quick reload on a nice summer day at the range, manual dexterity suffers a bit when your fingers are cold, all you can hear is your heartbeat, and you're neck-deep in thorny pucker-brush within twelve feet of a dangerous animal that you can't see, hear, or smell. In this case, a nice bolt gun, which requires less complex manipulation, would be my choice (if a napalm airstrike were out of the question).
I suppose I compensate for the lack of a fast follow-up shot by choosing larger-than-typical calibers for the task at hand. For hunting less threatening game, this has served me well in my limited experience. I do get teased a bit about overkill, but I'm red/green color blind and a lousy tracker. I'd prefer that the animal just drop and call it a day. So far, so good.
There is a great deal of satisfaction in hunting with a single-shot, particularly with carefully crafted handloads. Moreover, the ballistic "handicap" becomes a psychological advantage. I remember one particular jackrabbit hunt with several other fellows, one of whom was shooting an SKS with a 30-round mag. I was shooting a Winchester 1885 Low-Wall .22lr. I rolled far more rabbits, even with the "handicap" of an underpowered, obsolete rifle.
Using good judgment, of course, is paramount. There is no substitute for getting as close as possible, knowing your range and trajectory, and only taking high-confidence, responsible shots. The magnificent animal in your sights deserves that much.