The story
Modelled on the earlier foot-in-the-door technique, with the idea of having the door slammed in one's face when the original request is refused.
Etymology from Wiktionary, CC BY-SA 4.0.
door-in-the-face technique
Today it means: A compliance strategy in which an initial, large request is made with the expectation that it will be refused, followed by a second, smaller request that is the actual target. Full definition →