“We Didn’t Start the Fire” is not a complex song. There is a standard four-chord progression repeated in all the verses. The chorus provides the closest thing to a melodic hook, but even this sits on the same basic chords.
The song’s simplicity may be one of the reasons Joel has never been all that pleased with it. "It's terrible musically,” he has said. “It's like a mosquito buzzing around your head." Nor does he believe that it represents him as an artist. He has always been proud of the diverse influences he has brought to his music, such as classical, jazz, doo-wop, R&B, rock. This four-chord mosquito doesn’t really reflect this sort of breadth or complexity, though. As he says, the song doesn’t "really define me as well as album songs that probably don't get played.”
Part of the explanation may be that Joel reversed his usual creative process in writing this song. He was moved by the need to rebut what he perceived as a criticism of his generation, so he wrote the lyrics first; the music came later. In fact, he just borrowed a chord scheme that he had worked out for another song he had in process.