It is often said that dragons shed scales. These scales are thought to have magical properties. (Whether they actually do or not isn't as important as the perception that they do.) This grants them an economic value. Where there is an economic value, there will be someone
foolish brave enough to try to go after them. But of course, they're smart enough to not go after them personally. Nope. they instead hire people more foolish more brave than themselves, to get the scales for them.
Chicken eggs are so common. Tasty, sure. But have you ever eaten the extremely rare Omelette du Draco Ovum? No? Tsk. So sad. Well, obviously, only the most wealthy people can afford to dine on dragon egg omelettes. Some day, perhaps, you can afford the real thing. And no, ostrich eggs are no substitute. Sigh.
The bones of dead dragons are valuable, too. Ground up, they're said to be potent magic. Whole, they ward off evil spells. Ivory may be cheaper, but dragon bones are the real deal. Ivory just can't compare.
Teeth and claws
The sharp teeth or claws of a dragon make excellent daggers to impress your friends. Sure, they aren't quite as strong as good steel, but they're better than bad steel. And they don't rust. And dragons shed them. So it's not like you have to kill a dragon to get one. You just have to find one in the dragon's nest, before the dragon finds you.
Dragon meat never rots. No one knows why; magic, obviously. And unlike common pig or cow meat, it is quite spicy. A rare delicacy, then, to eat dragon meat. And a favored food to take on the road, since it isn't salty and never spoils. If you can get some, it makes the best stews, the best travel rations. Expensive, though. And don't be fooled by venison cured with pepper or other spices. That crap will rot once you're a week into your journey -- and away from the merchant.
A few rare fools think that they are dragon-marked. These idiots strike out to find baby dragons. They think they can tame the dragons. For every thousand or so who go out attempting this, maybe one succeeds. The tamed dragons aren't really worth the risks, to be honest. They don't grow as large. They usually don't learn to fly. And even if they do, they're never tame enough to safely bear a rider. They don't mate in captivity. But they are impressive.
Dragon dung burns like nothing you've ever encountered. Where do you think the Chinese got the recipe for black powder? They copied dragon dung. Poorly, I might add. Dragon dung is more powerful than TNT, pound for pound. Quite valuable in wartime.
Rite of passage
Maybe stealing a scale or lost tooth from a dragon nest is a rite of passage for warriors. To prove your a man and ready to begin your journey as a soldier. Or maybe it is a spiritual quest taken by certain monks of a select order.
Trial by dragon
Instead of a trial by combat, prove your innocence of [whatever crime you allegedly committed] by raiding a dragon nest. Come back with one dragon tooth or claw and one dragon scale, and you'll be innocent. If the dragon eats you, well, then we'll know you were guilty, you dirty criminal.