After Brian Robinson and Squeaky, I don't see how Lone Wolf says it CAN'T be done. It probably won't be done. But sure it CAN.
Thousands of Runners run 100 miles a day on potatoes, and drinks and soup. These foods can be lightweight. Throw in a fishing rod and a wildcrafter in springtime with edible plants abundant. No need to carry water or fuel on the AT (using a zip stove or hobo stove) and packweight comes way down.
I've seen Hindu pilgrims in Nepal going a 1000 miles with nothing but a shawl. sure they had handouts but i've also seen some excellent yogis on the AT. Bring in a dog or two as pets which can be eaten and a few other things your imagination can surely come up with and it isn't so hard. (is dumpster diving allowed?) There's no hurry anyway.
The only real hurdle i see is the person. Who would want to? probably not anyone i know. But make it a worthwhile challenge (say a million dollars or a "never have to work again" reward, and you'll have lots of people trying.