Now, there is growing concern among parents, donors and medical experts about potential negative consequences of having so many children fathered by the same donors, including the possibility that genes for rare diseases could be spread more widely through the population. Some experts are even calling attention to the increased odds of accidental incest between half sisters and half brothers, who often live close to one another.

“My daughter knows her donor’s number for this very reason,” said the mother of a teenager conceived via sperm donation in California who asked that her name be withheld to protect her daughter’s privacy. “She’s been in school with numerous kids who were born through donors. She’s had crushes on boys who are donor children. It’s become part of sex education” for her.

Critics say that fertility clinics and sperm banks are earning huge profits by allowing too many children to be conceived with sperm from popular donors, and that families should be given more information on the health of donors and the children conceived with their sperm. They are also calling for legal limits on the number of children conceived using the same donor’s sperm and a re-examination of the anonymity that cloaks many donors.

“We have more rules that go into place when you buy a used car than when you buy sperm,” said Debora L. Spar, president of Barnard College and author of “The Baby Business: How Money, Science and Politics Drive the Commerce of Conception.” “It’s very clear that the dealer can’t sell you a lemon, and there’s information about the history of the car. There are no such rules in the fertility industry right now.”

Although other countries, including Britain, France and Sweden, limit how many children a sperm donor can father, there is no such limit in the United States. There are only guidelines issued by the American Society for Reproductive Medicine, a professional group that recommends restricting conceptions by individual donors to 25 births per population of 800,000.