The infamous Dutch art dealer Michel van Rijn paid a man named Aydin Dikman to loot many ancient churches in Cyprus after the occupation by Turkish forces in 1974. Two of the most noteworthy artifacts later recovered were the 13th-century Byzantine dome and apse frescoes that had been stolen from a remote rural church in the Cypriot town of Lysi.

Dikman paid some locals to do the theft. The men were shown large B&W photos of the dome and apse frescos. An exacto knife and ruler had been used to scratch lines onto the photos to show the men how to cut up the frescos in order to remove them while doing as little damage possible to the figures of Jesus, Mary, and angels (otherwise the thieves might have chopped right through an angel's face). The thieves took a chainsaw and hacked the frescos from the domed ceilings of the church and then had the 38 pieces shipped to Munich, Germany.

The fresco pieces were later offered for sale to Dominique de Menil, a noted collector and art patron. The dealer claimed they'd been found in a private home in Anatolia, Turkey, however Dominique became suspicious and tricked the dealer into letting her photograph the fresco chunks. It took a year, but Dominque's art researchers managed to trace the stolen frescos back to Cypress where Dominique talked the archbishop and government officials into allowing her to purchase the stolen pieces, bring them back to her museum in Houston, and restore them. It cost her a couple million dollars, but she did it.

There have been a couple of books and a New Yorker article written about the Byzantine Frescos affair and Dominique's involvement in saving and restoring the frescos which were later displayed in Houston for 17 years and recently shipped back to Cypress.

Here's the wiki page about the Byzantine Chapel Frescos: https://en.wikipedia.org/wiki/Byzantine_Fresco_Chapel

Here's a few online articles: http://www.npr.org/2012/03/14/148602378/800-year-old-frescoes-headed-home-to-cyprus http://archive.archaeology.org/online/features/cyprus/

King Farouk of Egypt is widely rumored to have been an excellent pickpocket, and to have stolen, among many other things, the ceremonial sword and adornments from the corpse of the Reza Shah (the exiled former Shah of Iran's body was sent to Egypt to be buried, as it wasn't yet allowed back into Iran) and the watch of Winston Churchill.

These stories, however, seem difficult if not impossible to confirm. I've only seen it reported endlessly in cited popular history. I spent the last little bit searching around Google books--all I could confirm is that the rumors at least mentioned in his biographies, like Hugh McLeaves's The last Pharaoh: the ten faces of Farouk and Barrie St. Clair McBride's Farouk of Egypt, a biography, but Google Books' snippet view did not allow me to investigate further. This seems like exactly the sort of rumor that a new regime would concoct to discreet the old regime, however, and so without an clear eye-witness testimony or other primary sources, I can only say that this was widely rumored, not fully documented. The lack of any citation or quote about his thievery makes me suspect that this is something that only appears after he is deposed, perhaps at the instigation of the new regime which deposed him. Maybe someone knows more about modern Egypt and can give more context to these claims. Additionally, it seems like even in this rumor he's not usually commissioning the theft of specific items that he wants, but taking the items himself simply as opportunity arises, so again, neither the level evidence nor the exact pattern you're looking for. Part of the rumor is that he paid skilled pickpockets not to steal items for him, but to teach him how to steal.

Edit: it occurs to me that Churchill's ample biographies might have better documentation for this incident. It would have happened in 1942 or 1943 in Egypt (he visited Egypt at least twice in this period). Apparently, according to rumor at least, Churchill noticed the watch--a family heirloom--was missing, Churchill demanded it back, and Farouk was able to locate it quickly for him. That even is even reported in this 1970 gossipy newspaper account of Farouk's life. Anyway, I wonder if there's a Churchill enthusiast who can provide a more satisfying source for this incident.

To follow up on this, and perhaps this should be a separate thread, what about the "Nathan Drake" version of this question? Has there ever been, in modern times, a very wealthy individual or group, who has hired a group of specialists to find and raid a historical/archaeological site and abscond with artifacts for their patron's personal collection?

According to modern books on art theft e.g. FBI agent Robert K. Wittman's memoir and journalistic research into famous modern art heists, the answer is probably not, at least as far as art crime goes. While there appears to be evidence that there are criminal gangs who specialize in dealing with stolen art, it's extremely unlikely that their operations are being bankrolled by a secretive, wealthy benefactor who hides away his stolen treasures for private use. There is one book called Stolen Masterpiece Tracker by a retired FBI agent named Thomas McShane which claims that almost every art theft ever is absolutely ordered by some rich person who hires a team of criminals, however he offers absolutely no evidence to support this and quite frankly the rest of the book makes him sound like a bit of a loony old man so I can't say I trust what he says (not a terrible read if you're really interested in the topic though).

The boring truth is that, as far as we can tell, most art is stolen as a type of black market collateral or bearer bond. Valuable paintings and other artworks are often bartered for drugs, held as collateral on loans between groups, and sometimes used like retirement funds for career criminals. I can't recall which exact painting it was, but a few years ago I was reading a news article, and one of the works listed in Museum of the Missing was found practically by accident when police raided an apartment looking for a suitcase of drugs, only to find that the case had already been exchanged for the stolen painting.

The laws in various countries for the statute of limitations and ownership also make using pieces as long term investments reasonably practical. If a thief can steal a painting and successfully hide it from law enforcement for the period outlined by the law, they can legally claim themselves as the rightful owners of that piece, even if they were convicted and served time for its theft. This often ends up being a few decades but nonetheless, for someone who has no legal money to invest it's a tempting prospect, especially when the punishments for such crimes are relatively lenient.

Other times it's the thieves themselves who want to keep the artwork for their personal collections. A recent example is a French man (sorry for the wiki link but I can't remember which books he was listed in) who simply liked the art so much that he wanted it for himself. There was a similar case investigated by the FBI of a man in New York a few years back who liked to use his stolen art to help him seduce the young men he brought home.

tl;dr As far as art crime goes, there are several motives for thefts, but as far as we can tell none of them involve a greedy billionaire hiring a gang to provide him with his own secret room of treasures. It's certainly possible that statistically there are one or two people like that in the world, but even if this were the case it would only account for a very tiny percentage of art thefts. In reality it's mostly just regular people being greedy.

If you're interested in learning about art crime I highly suggest starting with The Rescue Artist by Edward Dolnick. He's a good writer and is very thorough in his research. Other books I've enjoyed in addition to the ones I already linked are Provenance, The Forger's Spell, and The Gardener Heist. There are plenty of other great books on thefts throughout the more modern era, such as The Napoleon of Crime and Flawless that aren't strictly art related but still good reads.

*Sorry I don't list more specific citations, but it's been a while since I've read most of these and it's hard to remember which information came from which books or was shared among several books.

Edit: typo