Naemanson, There are a few songs that I feel I can put over a little better unaccompanied. Some that come to mind are
Jack Monore
Jolly Grog
A Canadian Boat Song

I started doing Jack Monroe unaccompanied because I thought it had such a compelling, sparse melody that adding anything would take away. And besides, the recording I learned it from was made in the hills around Galax, VA by somebody from the LOC who recorded Crocket Ward and his wife Perline. Mountain music may be equalled, but it would be hard to improve on the Ward family.

As for Jolly Grog when you get the audience roaring along with you, what's the point of bringing in guitars, or mandolins and such? IMHO sea chanties are always best unaccompanied.

And then there's A Canadian Boat Song. The melody for this is a traditional French Canadian voyageur song originally titled Dans Mon Chemin. Thomas Moore learned it while on vacation in the US and Canada from some boatmen. He took the melody and wrote a separate set of lyrics for it, and had quite a hit back in 1803. It is fun to sing this a capella in three part harmony.