Catch-22 (n.)
from the title of Joseph Heller's 1961 novel. In widespread use only after release of the movie based on the book in 1970. The "catch" is that a bomber pilot is insane if he flies combat missions without asking to be relieved from duty, and is thus eligible to be relieved from duty. But if he asks to be relieved from duty, that means he's sane and has to keep flying.
There was only one catch and that was Catch-22, which specified that a concern for one's safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions. Orr would be crazy to fly more missions and sane if he didn't, but if he was sane he had to fly them. If he flew them he was crazy and didn't have to; but if he didn't want to he was sane and had to. Yossarian was moved very deeply by the absolute simplicity of this clause of Catch-22 and let out a respectful whistle.

"That's some catch, that Catch-22," he observed.

"It's the best there is," Doc Daneeka agreed.
See catch (n.).