Getting back on topic, if we're looking for a royal personage who was a victim of inbreeding depression the clearest candidate is Carlos II (1661-1700), the last Habsburg king of Spain. Most history books will tell you his deficiencies were the result of excessive inbreeding, but let's look at the evidence for ourselves.
How inbred was he?
That Carlos was the product of inbreeding is undeniable. Since the days of his ancestor Charles V (1500-1558, aka Carlos I of Spain), the Habsburgs had been in the habit of marrying members of the family's Spanish branch to members of the Austrian branch. Oh yeah, in their heyday the Habsburgs controlled not only the Spanish and Austrian Empires but also the low countries (Benelux), Portugal, southern Italy (Naples and Sicily), Milan and Burgundy.
Have a look at Carlos II's family tree. His mother and father were niece and uncle. Granted, this was sort of an accident of history given that this was Philip IV's second marriage, and it just so happened that none of the sons of his first wife, Elisabeth of Bourbon, lived long enough to accede to the throne.
A couple of other fun facts indicative of Carlos' pedigree collapse:
(1) Carlos was descended from his great great great grandparents Juana la Loca (1479-1555) and Philip the Handsome (1478-1506, Felipe el Hermoso) through three of their children.
(2) whereas someone whose ascendants (i.e. direct ancestors) were all unrelated to their spouses would have 32 great great great grandparents, Carlos only had 14.
How messed up was he?
Carlos' physical and mental deficiencies earned him the nickname "El Hechizado" (The Hexed). First off, there was his debilitating overbite. Prominent chins ran in the Habsburg family (check out the portraits of Charles V or of Carlos' father Philip IV), but with Carlos II the condition was so bad that he had problems chewing. It's also said that it was difficult to understand Carlos when he spoke and that he frequently drooled. To me this seems analogous to those overly purebred Persian cats whose faces are so flat that they have respiratory problems and their tear ducts get blocked up.
This, however, was the least of Carlos' problems. He was feeble and sickly, prone to high fevers which kept him confined to his bed. It's also said that he suffered from seizures, that he invariably vomited on carriage rides due to chronic motion sickness, and that his eyes oozed liquid in open air. In addition to his physical shortcomings, Carlos II was purportedly dim-witted and he was left basically uneducated. The classic 1911 Encyclopedia Britannica claims "there was no room in his nearly imbecile mind for more than childish superstition, insane pride of birth, and an interest in court etiquette." Due to his weak health, he remained mostly idle while his mother, his bastard brother Don Juan José of Austria, and his advisors made most of the decisions affecting the Empire, with Carlos' most vigorous activity consisting of the occasional hunt.
Although he was married twice Carlos II fathered no children, and it is assumed that he was infertile if not actually impotent. His reign as king was largely disastrous and his inability to produce an heir led to the War of Spanish Succession.
Was this inbreeding depression?
As I said above, Carlos' mandibular prognasthism sounds like a heriditary physical trait which became exacerbated to the point of becoming a deformity due to inbreeding. Moreover, his sickliness and sexual dysfunction appear analogous to immunity and reproduction problems which have been documented in highly inbred lab rats. Therefore, I think it seems very likely that Carlos' problems did indeed stem from inbreeding depression.
Interestingly enough, Carlos' sister the Infanta Margarita Teresa (1651-1673, who as a little girl was the subject of Velázquez's masterpiece Las Meninas) was not affected by the same problems as him. She was lively and beautiful although, judging by her portrait, she looks like she had a strong chin. Margarita was married to the Emperor Leopold I (her uncle and first cousin) who was himself pretty inbred and had an abnormal underbite although he was otherwise fit and intelligent. They had six children (only one of whom survived to adulthood) before Margarita died at the age of 21.
Images: portrait of Carlos II of Spain by Juan Carreno de la Miranda is located in the Kunsthistorisches Museum of Vienna; Ancestry of King Charles II of Spain found in Wikipedia Commons.