Dave Edmunds, the fantastic Welsh guitarist, singer, and producer (The Everly Brothers, The Fabulous Thunderbirds, The Stray Cats, his and Nick Lowe’s Group Rockpile, many others). His first two albums Rockpile (album title, not the Group) and Subtle As A Flying Mallet are mostly all he playing and singing every note. My favorite living musician.
Emitt Rhodes, whose self-titled debut album is he alone playing every instrument and singing every note of the all self-written songs. Better than the released-around-the-same-time debut solo album by my next nominee.
Paul McCartney, obvious but still worthy.
I think, whoever the musician might be, one man cannot be a band. I suspect even Paganini who is generally considered the best musician of all times, though not at all the best composer. You need at least two people. The two best duos I heard are Lisa Gerrard/Brendan Perry and John McLaughlin/Paco de Lucia.
I once saw the late great Michael Hedges play "Low Spark of High Healed Boys" on his acoustic guitar. It sounded like several people playing. Todd Rundgren plays all of the instruments on several of his albums Moby has a great time in the studio by himself. Basically, for better or worse, technology allows many artists to perform by themselves and sound like a full band.
Funny you should mention Todd Rundgren. On his record A Capella, every sound is Todd's voice (often electrically manipulated). It's kind of an ultimate "one man band" in that he needs zero strings (or keys, or anything else).
That album also contains one of IMO the most beatiful ballads of the pop/rock era - "Pretending to Care". Ironically, the version on A Capella isn't my favorite, but it's certainly an ambitious take on the song.
I just heard Marshall Crenshaw's fantastic version of Dylan's "My Back Pages" on You Tube. You may have heard The Byrds excellent hit version, but I like Marshall's version, found on the Bleecker Street various artists album, even more. And that reminded me that he is a multi-instrumentalist, playing and singing every part on his super-fine #447 album.
My favorite one man band is a buddy of mine - Rob Armus.
i especially enjoy his renditions of folk/blues classics - check out a sample of his one man efforts