Most Influential American Musician of All Time?

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.

Armstrong is the first to come to mind, although you could argue that Sidney Bechet trumps him.

IMHO Louis Jordan presaged rock n roll and Chuck Berry perfected it - both were highly influential in that regard.

Tvad made a good call on Robert Johnson, too. BB King influenced a lot of folks as well.

Lou Reed and arguably Brian Eno have had enormous influence on pop music post 1970.

For better or worse, Grandmaster Flash is among the most influential recent figures.

Lots of arguments for post war jazz giants, too, but I'll let better informed folks fight that fight.

All in, a very interesting question - and I'd probably end up back where you started - Armstrong.

Marty

the most important...armstrong. the most influential...presley

can't argue too much with Satchmo or Miles
a few others to consider

Duke Ellington
Jaco Pastorius
Charles Mingus
John Coltrane