I could make a list, but Zaikesman had it covered way back in 2002. His included some names I myself wouldn’t, but just about every one I would is there. I would not, however, list "Bob Dylan & The Band" as an integrated unit, but rather The Band on their own. He needed them more than they needed him!
Also from ’02, Trich’s mention of The Chocolate Watchband greatly amuses me! Growing up in Cupertino/SanJose/The Santa Clara Valley, I saw them live many, many times in ’65-6-7. I was actually at their very first public appearance, in ’65. They were a Garage Band, performing covers only, including songs by The Stones, The Animals, The Kinks, The Yardbirds, The Who, etc. When they got a record deal and traveled to Los Angeles to record, they threw together some "songs" for their three terrible albums. They are best known for being the Group appearing in the Roger Corman teen-exploitation movie, Riot On Sunset Strip.
In 1968, Watchband guitarist Mark Lumas and drummer Gary Andrijesivich (in the Cupertino High Marching Band and Orchestra while I was a student there) formed a new Group, The Electric Tingle Guild, who were much, much better. A 4-piece performing their own, well-written, songs, they were in fact good enough to include on this list.