I know it might sound a bit strange to ask, but what if Woodrow Wilson was elected President in either 1912 or 1916? He ran twice and both times he lost to Theodore Roosevelt. TR was of course insanely popular, but their is a way to get him elected.
What most people don't know is that; before the election, Incumbant President Taft was very close to seeking a second term. It was only after he and TR had a in depth discussion that he decided to opt-out of another term. This is what lead to TR getting the nomination and the Presidency again. In fact TR later said that even if Taft wouldn't have agreed to step aside, he still would have run for President, even if it had to be for a thrid party.
so, in the senario I purpose, Taft gets the nomination again, and TR makes his own third party, and thus splits the republican vote, allowing Woodrow Wilson to be elected President.
Now, how would a Wilson Presidency have turned out? What would America's role in World War One be like under Wilson? What would Wilson's domestic policy be?