Who's the most influential US politician of the 20th century who wasn't president?

I’d say just about everyone who was Secretary of State, primarily Dulles and Kissinger. Plus Eleanor Roosevelt and Martin Luther King (though they weren’t, strictly speaking, politicians).

a second vote for Sam Rayburn.

What’s a politician? Good question. I kind of like the broadness of Wikipedia’s definition:

So MLK and Eleanor Roosevelt would seem to qualify as people “who have the power to galvanize public opinion.” The category I was waffling on was justices, specifically Supreme Court Justices, and this definition also seems like it would include them.

Kissinger is a great answer, and one I should have thought of when I was writing the OP.

Daley and Huey Long inspire a question I might spin off into its own thread: the most influential politicians who didn’t hold national office. Though Long was a senator, even if that wasn’t the position he was most famous for.

George Wallace comes to mind.