Jeff Beck
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England’s favourite all-time flash guitar hero, Beck led the group bearing his name (The Jeff Beck Group), which was the nest for Ron Wood between Ronnie leaving the Birds and joining the Faces. Beck was also a serious contender in the Stones’ “Great Guitarists Hunt” after Mick Taylor’s departure, though rumor has always had it that Beck rejected the Stones, feeling that the rhythm section was not “funky” enough for his tastes. He did, however, record a few unreleased tracks in February 1975 with the Stones during the Rotterdam sessions. Later on, asked to participate in the New Barbarians tour in 1979, Beck uttered his favourite word, “No!”. Later on, Beck would record for Mick’s first two solo albums and other Stones-related assorted projects.

From MusicRadar:
Legendary guitarist (Doctor) Jeff Beck has told BBC 6 Music that he would still love to play with the Rolling Stones at some point.
Speaking to Matt Everitt, Beck responded to the question “Would you join them on stage?” by saying:
“Yeah, I would. I’ve just put my foot well into it, but I would, it would have to be a blues [number], or something that could show what I can do. I love them, who doesn’t? Even ‘her highness’ up front…”Beck’s clearly got The Stones on his mind, revealing earlier in the same interview that his one regret was not joining the band following Mick Taylor’s departure in 1974. “I would have loved to have been a Rolling Stone,” Beck told the Beeb. “But the thought would have been better than the act, I think – the fantasy of it. I don’t think I would have lasted, number one. I don’t think, musically, they were on the same path.
“I wouldn’t have had ‘Blow By Blow’, which has kept me as a solo artist… I’d have just been a member of the Rolling Stones. The biggest band in the world – how bad is that? If you can be someone on your own, then that temptation’s there. I guess it’s the ego thing – to have your own name on the ticket.”