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.