Many of his subjects are contained in his new book "Photographic Memory." From Fred Astaire to Natalie Wood and Stravinsky to Sting, they are all in there.
"When I first started out, I didn't care about money," says Claxton. "I just wanted to get good pictures. I'd borrow a camera and saved my money up for film. All I loved is just shooting. I didn't think about anything else."
Claxton grew up in the sprawl of Los Angeles. His heroes weren't in Hollywood. They were in L.A.'s jazz clubs —exotic places he started sneaking into as a teenager.
"I would be with the musicians day and night, whenever I could, listening to their music mainly," recalls Claxton. "And of course, I had my camera."
It was a borrowed camera he had in the beginning, with just a single roll of film that had to go a long way. Along the way he made a lot of pictures and friends.
"I really chased after musicians," says Claxton. "They're the only celebrities I've ever chased after. Everybody else I've been hired to photograph. But jazz people, I really adored."
His idol was the saxophone player Charlie Parker, "Bird," as he was known. After a long night at L.A.'s Tiffany club, Claxton invited Parker home for breakfast when his parents were out of town, and later told them about it.
"I said, 'Mom, mom, mom, guess who was at our house? The world's greatest jazz musician.' And she said, 'Who?,' And I said, 'Charlie Parker.' She says, 'Oh, that's nice. Did you give him something to eat?'"
It was trumpet player Chet Baker, the lone white member of Charlie Parker's band, who caught Claxton's photographer's eye.
"He looked like a very, very pale white, angelic prize fighter," says Claxton. "He was tough and husky and strong. And a very, very pretty face with one tooth missing."
Claxton later did an entire book on Baker because what came out on film after that first shoot was stunning.
"When I went to the dark room and developed the pictures of the group, it was Chet Baker's face that came right out at me. I thought, 'My God, that's what photogenic means.'"
Claxton made an art form out of album covers — card board record containers that were like billboards for the music. Typically he'd have the artists pose on location, such as on a boat, the beach, or a merry-go-round. He created an image for the artists and a trademark look for the art form that became known as West coast jazz.
Claxton believes there's a reason he was so successful shooting jazz musicians.
"There's a big similarity, for me, between photography and jazz," he explains. "A musician has to really master his instrument, his 'axe,' as they call them. And photography, a photographer really masters his 'axe' with a camera. So, they're free enough that they can improvise. So, I feel like photography is jazz for the eye."
One of Claxton's favorite pictures was taken in Los Angeles's Echo Park District. Saxophonist Art Pepper was just out of prison after serving time for narcotics. He'd struggled with drugs his whole life. And it was this struggle that inspired Claxton to have Pepper walk up a steep hill and stop.
"He had his alto saxophone and he walked up and I clicked the shutter," says Claxton. "To me, that summed up the struggles he was. He was telling me about his whole life. It was all uphill, all the time."
Claxton's success with West coast jazz opened more doors for the young photographer.
"The jazz thing got bigger and bigger and bigger, then the pop people wanted me," says Claxton. "Frank Sinatra wanted me. And Peggy Lee, and a lot of the big recording stars. And then I got elevated to a little higher level."
Claxton easily segued into fashion photography, not surprisingly since he married Peggy Moffitt, a fashion model. She was fashion's "it" girl in the 1960s, displaying the colorful, outrageous fashions of Rudy Gernreich in photos by Claxton.
Moffitt says having Claxton behind the camera helped her in front of the lens.
"Most fashion photographers tell you what to do," she says. "[Claxton] respected me, and so did Rudy. So for me, I could do what I do well, which is respond to clothes, and perform in them."
One picture drew particular attention. It was Gernreich's topless bathing suit.
"I wanted it to be a fashion statement," says Moffitt. "I think it was prophetic. I think it was the moment that fashion did become modern. Not because people were wearing the topless bathing suit, but it affected clothes. It affected the way dresses were made."
Claxton says the times were wild for Moffitt, Gernreich and himself from 1964 to 1965.
"It was pretty avant-garde," he explains. "I think it put Rudy on the map, put him on the cover of Time magazine, put Peggy on the cover of Time magazine."
Claxton worked outside still photography for a time, turning his talented eye to commercials and television sitcoms as a director. But, the fast pace wasn't compatible with his easy-going style — a manner and reputation that opened doors to entertainment icons.
Claxton did a full book on Steve McQueen, the actor who never trusted anyone, especially still photographers. But, he was won over by a surprising gesture.
"One day I just put the camera, my still camera up and showed him how it worked, almost like you would do to a child or an aborigine," laughs Claxton. "I let him look through it, and I said, 'This is the best time [for a shoot]. You do this and you'll look wonderful.' And I said, 'Can you feel your light? Can you feel this happening to you?' And he got so involved, that he said, 'Yeah, I see what you mean. So, that's your technique.' And we started shooting."
Claxton says people sense that he is a nice guy that isn't trying to capture an ugly image of them.
"I like beauty, and I respect beauty," he says. "And I respect talent, so I like to get people looking the best possible way."
That respect was put to the test by the two faces of Judy Garland. At the time of Claxton's photo shoot with the actress, she was on pills and booze in her dressing room.
"It was hard for me to photograph her because it kind of hurt me to see her that way," says Claxton. "She didn't mind at all."
And onstage, two hours later: "All of a sudden, wham! She walked out there and she just became fantastic," says Claxton. "You'd never know it was the same person."
Claxon had a private moment with Ray Charles during a break in a recording session back in 1962.
"[Charles] said, 'Where are the timpani's? Show me what the timpani looks like,'" Claxon remembers. "And I led him over to the timpani and he touched them. He said, 'You know, I've never seen timpani's before.' And we did that with a lot of the instruments. And I photographed him each time. And it was a very, very touching thing to hear him say, 'I've never seen one of those before,' because touching it was for him like seeing it."
In 1964, when Barbara Streisand came in for her first fashion shoot, Claxton asked his wife to coach her.
"[Moffitt] would show it and then Barbara would imitate and do her version of it," says Claxton.
The Claxton-Moffitt collaboration has lasted longer than most in the entertainment capital.
"We just like each other and we're great friends with one another," says Moffitt. "I'm not going to say that we've had 43 years of whipped cream. There have been a few nuts in various pieces mixed up in there."
"A little peanut brittle in there, now and then," jokingly adds Claxton.
Claxton is still at work. Most days he is still the most famous person at the photo shoot. Camera lens and steady hand in concert, he improvises another little number — a little jazz for the eye.