Clinton goes to jail for her carelessness with classified data, and Trump goes to jail for Trump University, and the office of President is left vacant after Obama. Every bill is considered vetoed; if a bill can't overcome a veto, it's probably not one we need anyhow. With no new president anxious for the "greatness" that comes of being a "war president", maybe we could avoid wars in which we are not directly attacked. At any rate, it would lie with Congress to declare a war, you know, the way the Constitution says.
Of course, this won't happen. Some numskull would consider it a "long national nightmare" and act to prevent it. The real worry would be that Americans might find a government with no president preferable to one with a president.