"What songwriter's songs have appeared on albums by the Beatles, Dylan AND the Rolling Stones?
Why, Alabama's own Arthur Alexander. He is the writer of "Anna, Go To Him" (as well as "You Better Move On", "Sally Sue Brown", Elvis' "Burning Love" (co-writer) and bunches of other '60s/'70s songs -- also wrote, amazingly, Dr. Hook's schmaltzy "Sharing the Night Together" which I heard all the time on the radio as a kid). Paul McCartney has said in interviews that the Beatles tried mainly to sound like him.
He either washed up or voluntarily retired in the late '70s, and ended up in Cleveland driving a bus that carried disadvantaged children around from place to place. He survived long enough to be inducted into the Alabama Music Hall of Fame and had just started doing music again in the early '90s when he died at 53."
I found out about Mr. Alexander through the Beatles "Live at the BBC" recordings, including "Soldier of Love" which I think Pearl Jam also covered a few decades later.
Here's another one of Arthur's tunes that the Beatles covered...
Here's the Beatles' Version from "Live at the BBC":