Flow My Tears, the Policeman Said, by Philip K. Dick
This must be one of the odder titles in SF, but then this is Philip K. Dick we're talking about--weird is what he did. While I've read some of Dick's short fiction, I don't think I've read any full-length novels. Maybe.
Jason Taverner is one of the most famous celebrities in America--millions watch his show every week. He's made several record albums, he's in every issue of every gossip tabloid, everyone knows who he is. And one morning, Jason Taverner wakes up with no identity. No one remembers him. There is no record of his birth or his fingerprints or anything; and in the fascist American state of 1988, a person can't even walk a few blocks without proof of ID. How can Taverner survive without getting killed or sent off to a labor camp?
The solution to Taverner's sudden loss of identity--partial as it is--only comes at the end, and is much, much weirder than I expected. The whole story is bizarre, but it sure is interesting. I would have liked more explanation of the world, especially why students live as outlaws, underground in kibbutzes, and aren't allowed into wider society, but Dick was a pro and knew better than to explain everything to death.
Also I think this premise must have informed the creation of UPN's 1990s show, Nowhere Man. Does anyone remember that? My husband and I used to watch it when we were first married. The end was pretty terrible, if I recall, but we enjoyed it a lot.