As best as I can tell, the last time we had a presidential election without an incumbent President or Vice President was 1920. The unpopular and, at that point, invalid Woodrow Wilson had just finished his 2nd term. Wilson's Vice President was a fellow named Thomas Riley Marshall. Marshall had served two full terms as Vice President, although he and Wilson didn't get along particularly well. Marshall was a quick wit who was known to greet visitors on the White House tour by asking them to "be kind enough to throw peanuts at me". Marshall didn't seek election but went back to Indiana and resumed his law practice.
There you go. Enjoy the primaries and I'll get back to Joel next time since that's why my 3 loyal readers are here anyway!