Incest

"Brothers and sisters? Where I come from that stain is very difficult to wash out."

- Olenna Tyrell to Tywin Lannister

Incest, the act of engaging in sexual intercourse with one's direct family member (siblings, parents, offspring) is a great cultural taboo both in the Seven Kingdoms and the lands beyond the Wall.

Alongside kinslaying and the violation of guest right, incest is proscribed by every major religion in Westeros. Children born of incest are considered abominations

Known instances of incest

 * The members of House Targaryen, like their ancestors of the Valyrian Freehold, often married brother to sister to keep their bloodline pure. However, generations of such heavy inbreeding increasingly produced insanity on some of them. Despite the Faith of the Seven proscribes incest, it turned a blind eye to the practice of the royal family.
 * Queen Cersei and her brother Ser Jaime Lannister have carried an illicit romance since childhood. Cersei's children: Joffrey, Myrcella, and Tommen, are all born of their affair.
 * The wildling Craster takes his daughters as wives and does the same on the daughters he sires on them.

In the books
In the A Song of Ice and Fire novels, the Targaryens continued to incestuously marry brother to sister "to keep the bloodlines pure" as had their ancestors in the Valyrian Freehold. For a time, at least, they also continued to practice polygamous marriages. Aegon I Targaryen ran into some opposition after the War of Conquest because of this, as he was married to both of his sisters, Visenya and Rhaenys (no subsequent Targaryen king is mentioned to have continued practicing polygamous marriage). Aegon I was a strong ruler, but after he finally died and was succeeded by his son Aenys, the Faith of the Seven rose up in rebellion against such abomination, using their military orders known as the Faith Militant. They encouraged many lords and commons throughout Westeros to join the rebellion. Aenys was a weakling, but Aegon's younger son Maegor was a cruel and ruthless leader who served as his Hand of the King. The Targaryens couldn't denounce their incestuous practices by this point, because it would mean Aegon's heirs were illegitimate. Maegor brutally crushed the armies of the rebellion whenever they marched against him, but they continued to resist for years. Aenys died after ineffectually ruling for five years, and Maegor seized the throne ahead of Aenys' son Jaehaerys, and continued to persecute the Faith. Maegor himself died without issue after ruling for six years, and was succeeded by Aenys' son Jaehaerys. A wise and capable negotiator, King Jaehaerys I ended the eleven-year long Faith Militant Uprising by offering the Faith's leadership a compromise: they didn't have to praise the incestuous marriage practices of the Targaryens, they simply had to acknowledge it. The Faith had been defeated militarily, but Jaehaerys' terms were generous (including a complete grant of amnesty so long as the Faith Militant was disbanded), so they accepted the offer. Ever since, the Faith continued to officially consider brother-sister incest to be an abomination, but treated the Targaryens as exceptional due to their royal status. Meanwhile, the Targaryens continued to wed brother to sister for three hundred years whenever possible, though in some generations in which no daughters were born they did have to marry outside the family.

Possibly as a result of this, incest prohibitions in the form of laws of Consanguinity do not appear to be as strict in the Seven Kingdoms as they were in the real-life Middle Ages. The medieval Catholic Church initially imposed restrictions on a person marrying their relatives to the seventh degree, but later at the Fourth Lateran Council in 1215 changed this to within four degrees. This does not refer to "seventh cousins" or "fourth cousins". The old Roman method for reckoning degrees of relationship was to count "acts of generation", thus a person is one degree away from their parents, two degrees away from their grandparents, three degrees away from their uncles, and four degrees away from their first cousins. Thus it was forbidden to marry blood relations up to a second cousin once removed (or a first cousin thrice removed). After the barbarian invasions, however, a new complication arose, in that the Germanic system of reckoning blood relatives was different from the Roman system. The Germanic system was based on how many parental generations back two people possessed a common ancestor: siblings share common parents so they are only one degree removed (instead of two degrees as under the Roman method), and first cousins are two degrees removed from each other, as their closest shared relatives are their grandparents (instead of being four degrees away as under Roman law). The problem that arose from this change is that while the method for reckoning degrees of blood relationship had changed, the strict wording of the Church's laws restricting marriage "within seven degrees" was not updated to take into account the fact that the definition of what a "degree" was had changed. As a result, everything up to marriage between sixth cousins was considered forbidden - which was practically impossible to avoid.

That being said, the Catholic Church was not frequently able or even willing to enforce such overly strict interpretations of the prohibition. It usually only enforced consanginuity laws when it wanted to apply political pressure on a secular lord seeking a marriage, and conversely, would often grant dispensation to a lord seeking to marry his third cousin if he was on good terms with the Church (i.e. gave them generous land donations). By the thirteenth century, however, it was decided to finally update the official Church laws on incest to reflect the fact that they were using the new Germanic method of reckoning blood relatives: thus the Fourth Lateran Council of 1215 officially lowered the number of degrees in which marriage is forbidden from seven to four. This does not mean that long-held prohibitions on incest were suddenly relaxed, but rather that the laws and method of reckoning were finally brought closer back to their original alignment (forbidding marriage between what we would call "second cousins").