Those singers you enjoy,....that don't really have a great voice?

Frankie Beverly (and MAZE)

Kristin Asbjornsen (great singer, i would say, but some would find her voice to be grating!)

I would add Ry Cooder, a great favorite of mine not blessed with golden pipes.  John Lennon was no bargain either.  Arlo Guthrie, Kieth Relf of the Yardbirds, 

Moody Blues

Bob Dylan

Louis Armstrong

Leonard Cohen

All great singers in their own way.

Of the big four mentioned above, Neil Young, Leonard Cohen and Dylan and Jerry, Neil and Jerry are the ones I can listen to throughout entire career. When Cohen and Dylan began to sound more like croaking frogs than human is day I couldn't take it anymore.

There is this one obscure female hillbilly singer from 1950's, can't recall her name for life of me, somewhat recent thread here mentioned her. Anyway, the voice is probably the single most unbearable voice I have ever heard!