Krispy Kreme made most of its rep on its physical locations where you could get a a fresh doughnut nearly right off the cook line, I think. The kind of basic glazed yeast doughnut that is your standard Krispy Kreme is a lot like french fries - they're AMAZING when they're hot, fresh out of the fryer/oven, and still warm enough to keep the glaze mostly liquid. But as soon as you start letting them cool they lose quality more rapidly than most foods. The glaze settles and gets weirdly thick in uneven places, the air pockets collapse and the doughnut goes from being this light puff of air with a pastry shell to getting just kind of.. well, doughy, and it's all around just not an inspiring experience any more. If you leave them too long they start staling in really obvious ways; if you have to get older cooled doughnuts off a display case, practically every other style of doughnut holds up better than the basic yeast-risen one.
Which is to say I don't think there's anything particularly outstanding about Krispy Kreme's recipe or their baking - they were just the chain that introduced a lot of people to properly fresh hot doughnuts with their 'FRESH HOT' sign gimmick letting the public know when a batch was up, rather than ones that inevitably lose their best qualities no matter how good the baker is unless you happen to know just when they get a tray done.