The Tennō, the Emperor of Japan, claims descent from the kami of the sun, Amaterasu Ōmikami. The current Tennō does not put any emphasis on that claim, but the Shinto establishment does regard it as important. This is also fairly well known outside Japan.
What is less well known is that this is not at all uncommon. The Sengë, the hereditary priests of Izumo Ōyashiro (also known as Izumo Taisha, in a different reading of the same characters), also claim descent from Amaterasu Ōmikami, through her second son rather than her eldest. The Amabë are the hereditary priests of Kono Jinja in northern Kyoto prefecture, and also claim descent from Amaterasu Ōmikami, through an older brother of the grandson who founded the imperial line. (The Imperial family is not special simply because they are descended from Amaterasu Ōmikami; when an Imperial princess married the heir to the Sengë family a couple of years ago, she still had to leave the Imperial family, even though her husband also claimed descent from the same kami.) The Aso family supplies the hereditary priests of Aso Jinja in Kyushu, and claims descent from the kami enshrined there. All of these families claim eighty generations or so, and over a thousand years of that pedigree is as securely established as such things can be. That doesn’t quite get you back to the kami, however.
Historically, there are quite a few examples of priestly families that claim or claimed descent from the kami they served. (The Sengë are unusual here, in that Izumo Ōyashiro enshrines Ōkuninushi no Ōkami, not Amaterasu Ōmikami. The legend is that Amaterasu Ōmikami promised that her descendants would serve him as part of the compensation for his giving up rule over Japan.) The first priest of Ōmiwa Jinja in Nara prefecture, for example, is described as a descendent of the kami there.
A particularly interesting example is the Nishitakatsuji, the hereditary priests of Dazaifu Tenmangū in northern Kyushu. They also claim descent from the kami of the jinja, and in this case their claim is almost certainly accurate.
The kami in question is Tenjin-sama, who is also Sugawara no Michizanë, an important historical figure of the late ninth and early tenth century. He was famed as a poet and scholar, and also as a successful politician who was ultimately pushed out of power by his rivals, and died in exile in Dazaifu. After a series of disasters, he was deified as the kami Tenjin to pacify him. (More details are available in my Patreon essay about the kami.) As he was a member of the nobility, records were kept of his descendants, and so there is no particular reason to doubt that these priests really are descendants of the kami they serve. (They are only up to forty generations or so.)
In a more recent example, a descendant of the Tokugawa family recently served as a priest at a Tōshōgū, a jinja enshrining Tokugawa Ieyasu. As Ieyasu only died in 1616, and his family ruled Japan until 1868, doubting his descent from the kami is on the level of a conspiracy theory.
Not all Shinto kami are mysterious figures from ancient myth, so some priests are descended from the kami they serve, and they can prove it.