Relative to the country at the time he served, who was the most liberal U.S. president in history? Most conservative?

This is more political science than history, but I think it's relevant to mention Stephen Skowronek. Skowronek divides presidents into four categories: reconstructive, preemptive, articulation, & disjunction. Reconstructive presidents overthrow the dominant regime and establish a new one that dominates politics for a significant amount of time. Franklin Roosevelt is the obvious example of a reconstructive President. I would argue that President Franklin Roosevelt did more than any other President to shift American politics to the left. Of course, that isn't your exact question. Presidents are created by their circumstances. Roosevelt was successful not only because he was a smart politician and he didn't shift the country leftward solely because he was a liberal. He benefited from the fact that the existing regime was falling apart and he came into office in a time of crisis.

Answering these questions are difficult. It's tough to know what a President's ideal America would look like. Presidents are restricted by the other branches of government, by politics, by institutions, and by their predecessors. President Reagan ran a government larger than FDR's, even if you ignore defense spending. Does that mean that Reagan was more liberal than FDR? Most would argue no, but it's impossible to escape the fact that the policies of former Presidents don't get overturned overnight once the next guy comes into office. Policies become accepted over time. Medicare wasn't popular like it is now at first. But you don't see any mainstream Republicans calling for the end of Medicaid. Is that an indicator that the country has shifted to the left economically? On the other hand, President Reagan shifted the tone of the conversation away from the New Deal/Great Society consensus of the government helping people. The Democrats got destroyed in three straight elections until Bill Clinton came along and shifted the Democratic Party toward the center. So maybe our politics have shifted to the right.