Updated October 4, 2019: removed link as it’s now a dead link.
Iceland is isolated and tends to have few non-natives move to the country. It also has a small population: hair shy of 350,000. They created an app to see if your potential date is too close to marry: https://www.cbc.ca/news/business/kissing-cousins-icelandic-app-warns-if-your-date-is-a-relative-1.1390256.
I bring this situation up because there are too many situations in the U. S., the U. K., and other countries where you really should make sure you aren’t related before you decide to have children with someone. A recent example is Accused Fertility Doctor Claims Law Protects Him from DNA Testing. The doctor probably didn’t only father the one child using his clinic. He most likely fathered a lot of children this way.
According to this article, a Dutch donor may have 1,000 children: https://www.bioedge.org/bioethics/dutch-man-may-have-fathered-1000-children/12844. These children are half-siblings to each other.
In a slightly different variation dating back to World War I through 1950, you have this case – https://www.spectator.co.uk/2013/08/a-secret-sperm-donor-service-in-post-first-world-war-london/.
Derek fathered 496 children between 1917 and 1950. He had three sons by his only marriage, and two further sons by his father’s mistress. Then there were four more sons left behind in Malaya. The rest of his progeny were conceived to patients of Helena Wright’s Knightsbridge and Notting Hill clinics.
In addition to donors, you have other cases where the parents aren’t who you think they are. Adoptions are a good start. I ran across several articles from around the world where siblings married. Here’s one case – https://www.dailymail.co.uk/news/article-507588/Shock-married-couple-discovered-twins-separated-birth.html (link no longer works, but was a hoax), but it’s happened more often than people realize.
The above scenarios are not the only reasons to DNA test. There will be cases of rape, infidelity, or other cases where the mother and/or father aren’t the ones listed in official or unofficial records.
There seems to be a trend these days to claim that close relatives having children is no big deal, but those experts ignore cases like this: http://www.digitaljournal.com/article/195535. It may be an extreme case, but it’s also a good reason why you shouldn’t have children with close relatives.