What have been the missions of some notable high-endurance aircraft -- ones that can stay airborne for several days. Obviously, high-endurance missions have refueling, food, and human factors concerns that have to be addressed; what have been the missions that necessitated high-endurance aircraft?
Historically long duration flights were simply a way to fly further. Before the jet age crossing oceans was a time consuming affair and required what some may consider a "high endurance" aircraft/flight. Keep in mind it took Charles Linbergh in the Spirit of St. Louis 33.5 hours to cross the Atlantic and this was done with no refueling.
On the scientific/record setting front
A variety of airframes have been used for setting world records arguably a stunt in some cases and others with the pursuit of scientific progress. The current flight endurance record is held by Robert Timm and John Cook at a whopping 64:22:19:05 (thats 64 days, 22 hours, 19 minutes and 5 seconds) in a Cessna 172 of all airplanes quite an endorsement for the endurance of the airframe/engine combo.
Solar Impulse a modern experimental solar aircraft that recently flew around the world regularly flew 15-22 hour legs due to its slow airspeed. In this case the mission effectively required long legs due to the aircrafts performance. This was a purpose built plane.
The Rutan Voyager was another purpose built aircraft that became the first airplane to circumnavigate the globe, non stop and with out refueling remaining aloft for 216 hours 03 minutes and 44 seconds.
On the operational front:
The Air Force operated Boeing VC-25 (often called "Air Force Once") is capable of aerial refueling and could remain aloft for quite some time and is built to be a mobile command center should the need arise for it.
The Boeing E-3 Sentry commonly known as AWACS provides aerial radar support and is capable of air to air refueling allowing it to stay aloft for well above its standard 8 hour endurance. I dont know a whole lot about its standard mission so Im not sure it ever flys multi-day missions.
There is a nice list of long range airliners here all of which have considerable endurance (but dont meet your multi-day requirement) and are all built (more or less) for servicing non stop routes from far locations.
Its worth noting that the general civilian use case of and airplane does not really lend to high endurance planes (multi day kinda stuff) in practice. Planes are generally for moving people and cargo in a timely manner and not really for taking off and just loitering for a while. The various militaries of the world simply move their aircraft around on aircraft carriers and launch them closer to where they are needed or fly high speed aircraft from farther away when needed. Airplanes also have the luxury (if permitted by laws and airspace regs) to move directly from point to point greatly reducing the distance traveled to get between locations over other methods of transit, again cutting down on travel time.
Some consider the space shuttle an aircraft and it was capable of multi day missions for the purpose of space exploration, satellite service and delivery and ISS transportation.