The problem with many of the known mules is that there may be only one in existence, like that 1959-D wheat reverse. That smells like fraud, either from outside the Mint or from within. The shadow of suspicion over that particular coin hasn't stopped it from selling for tens of thousands of dollars, though.
A more populous example is the Sacagawea/Quarter Dollar Mule.
Photo courtesy Heritage Auctions, used by permission
There are 11 of these known and the U.S. Mint apparently knew that it accidentally created several thousand, but thought that it had recovered and destroyed all of them. Yet, in 2000, one turned up in a 25-coin roll wrapped in a U.S. Mint paper wrapper.
There could be more. Sacagawea's don't circulate much. Most are still sitting in rolls and may never have been touched by human hands. If you were able to find one, it might sell for a lot of money: one sold for $155,250.00 in 2012!
More about the Sacagawea/Quarter Dollar Mule.
Some unique (and possibly fraudulent) mules:
1859 Two Headed Indian Cent
1995 Cent on dime planchet with dime reverse
1999 cent on cent planchet with dime reverse
Two tailed Washington Quarter
Clue for the second 2014 Silver Eagle Giveaway: He was a musical prodigy also!
Note: All my coins are in a safe deposit box. I keep nothing in my home.
Contests and giveaways: http://coins.aplawrence.com/2014/05/contests-and-giveaways.html