Occasionally, a budgie breeder might find a baby in a nest that starts out looking like a normal baby, but this changes as the baby starts to grow feathers. Commonly called a “feather duster” this type of budgie grows feathers that curl and continue to grow without stopping as a normal bird's feathers stop.
Usually, these birds do not live long. The reason for the short life seems to be that the bird can not get enough nourishment to support the constant feather growth as well as to keep his body alive. Providing a more nourishing diet may extend the life of the budgie, but unfortunately, he will never be able to fly or even to perch normally. The feathers also cover the bird's face and eyes, which make it impossible for him to see well and interferes with eating.
Some people clip the feathers, especially around the head, so the bird can see and eat as well as around the vent, so he does not constantly dirty himself when he eliminates.
Since budgie breeders do not like hatching babies that will only live a short time, most will not breed the same parent birds together again, even if most of the babies hatched are normal.
If it was not for the fact that these birds live extremely short lives, they would probably be quite popular as pets because of their unusual appearance.
I have never personally seen a feather duster, but found some pictures of some at feather dusters.
There are budgies that have slightly curled feathers, but are not feather dusters. These birds' feathers stop growing when they should stop and the birds have a normal lifespan. One of these is the crested budgie and as the name implies, the curled feathers are on the crest (or top) of the head.
There are also some that have a few, or even just a couple, of curled feathers on the back. This is a fairly new mutation and is not at this time very common as these birds are usually kept back by the breeder. The canary breeders have been breeding curly feathered canaries called frills for many years.