Most Influential American Musician of All Time?

I agree with Jeffery. I really can't think of anyone that comes close to his influence from a historical perspective although I'm sure some will say Miles Davis but not from a musician's standpoint IMO atlthough his music was certainly innovative and took jazz in a different direction. What is unique about Louis Armstrong is that he gave his instrument a voice that was unheard of up until that time and was built on by subsequent musicians. The major icon of jazz.

No Brainer - Robert Zimmerman

While I think that both of the above musicians are worth candidates, (Robert Johnson and Louis Armstrong), but since I listen to Rock mostly, I would nominate Bob Dylan.
(He is not really my favorite, but between all the musicians who do idolize him, and/or those musicians who emulate his style, I think he is worthy of consideration.

My two cents worth.