This is progressive storytelling with a twist! An introductory sentence is decided upon (or provided by the first storyteller), and then each person, in turn, adds another sentence to develop the story. The twist lies in the necessity for each new sentence to begin with the last word of the previous sentence. The only exception to this rule is to allow "a", "an" or "the" before the required word.
Before beginning, you might want to stipulate how many sentences the story must have, or how much time the story can take before the conclusion.