Some Beatle-related ones...
Jimi Hendrix lives and forms a band with Paul McCartney and Miles Davies
Some others...
Syd Barrett doesn't go insane
1. Wait, what? I could see some kind of Hendrix and Davis collaboration as Bitches Brew indicates the latter was interested in exploring a more rock influenced Jazz sound. But how would Paul McCartney fit into that mix?
2. From everything I've read and heard, avoiding Syd Barrett's meltdown would be hard, as he seems to have had some kind of underlying condition for most of his life, though if his family is to be believed, it wasn't Schizophrenia. Not sure precisely what was wrong with Barrett because I've heard arguments in favor of various diagnoses, but whatever the condition was, it may not be possible to avoid entirely.
Here are two divergences which could alter the late Beatles period pretty substantially. Not sure how precisely to pull them off, and they both require some sort of earlier divergence to justify as you might need to solve the management puzzle, but anyway what if
1. Paul McCartney had agreed to release Cold Turkey as a Beatles single as John Lennon originally offered?
2. Let it Be had met with McCartney's satisfaction somehow?
The result? Lennon is still ambivalent about the Beatles into the end of 1969, and McCartney doesn't announce his departure from the group when he releases his first solo album. Later on the Beatles continue to occasionally record together in the 1970's. Everyone in the group has said they could have continued on for at least a little while longer than they did even if a split eventually was inevitable.