I actually knew zero Beatles songs for quite a while....Aerosmith's cover of "Come Together" along with "Yellow Submarine" from my choir class were the first I got to know of them....and that was in fifth and seventh grade respectively.....I honestly hadn't heard more than five Beatles songs until 1 came out and my friend played it at a party once....and even then, I never heard them again until I purchased Abbey Road. So I guess it is possible for someone to listen to lots of rock music and somehow avoid them all together, as I never even heard an actual Beatles studio track until the year 2000.
EDIT: On a side note, I did hear "Free as A Bird" around the time the Anthology came out, but that was about it.