“I think I would be in California,” she says. “Maybe I would be married? Maybe I would have kids? I would like to believe I would. I would have probably gone into some kind of science. I love animals.” She rubs Lorelei on her pink, white-haired belly. “Maybe I would have become a veterinarian.”

I ask Williams who she thinks she would be right now if she hadn’t ever picked up a tennis racket.

It’s early spring, and Williams is sitting on an outdoor couch near the pool at her handsome two-story home in Palm Beach Gardens, Florida. Chip, Williams’s energetic Yorkshire terrier, and Lorelei, her Maltese, roll on the cushion between us. Williams is dressed post-training casual: purple Nike headband, white top, shorts. Her feet are bare.

SERENA WILLIAMS does not consider herself a morning person—“Not at all,” she says with a distressed look—but if she could wake up one morning and not be a globally famous tennis champion, here’s what she would do: go to an amusement park. Or shop at the grocery store. “Ordinary things,” she says. “I’d go to the mall. I never go to the mall.”

There’s often an assumption with sports superstars that once they’ve ascended to the top, the hardest work ends. But the opposite is usually true. Whether it’s Steph Curry with his three-point jump shots or Tom Brady throwing spirals, the greatest seem to work harder once they’ve been recognized as great, as if that status is perpetually under attack. Williams is the same. Behind the scenes she is known as one of the hardest workers in the sport. “The number of hours is one thing, but [more] impressive is the effort,” says Mouratoglou. “She expects the best from herself.”

LAST YEAR, I WATCHED Williams stand in the hot afternoon sun in New York and hit serves for an hour and a half. Just serves, one after the next. She had won an early-round match at the U.S. Open, but her serve had been rocky, and the match wound up narrower than it should have been. Minutes after leaving the Arthur Ashe Stadium court, Williams walked out to the practice area with her coach, Patrick Mouratoglou, and began to serve. A crowd quickly gathered—Serena Williams! Williams kept serving. The crowd grew. Williams kept serving. And serving. After a while, the crowd got a little bored and began to thin. Williams kept serving and serving. She stayed until she felt she’d gotten it right.

She is very much still going. When she is playing at her best, she remains unstoppable. At an age when tennis players are often years retired, she is rewriting the books. Elite athletes who have breathed similar success cede to Serena’s greatness. “You can’t talk about women’s sports without first thinking of Serena Williams,” says Abby Wambach, the recently retired U.S. women’s soccer star who won two Olympic golds and a World Cup. To Wambach, Williams embodies “power, strength, beauty and confidence.”

“I’ve had people put me down because I didn’t look like them,” Williams said in a speech captured in her documentary, Serena, which arrived on Epix June 22. “I’ve had people look past me because of the color of my skin. I’ve had people overlook me because I was a woman.... I’m still going.”

Williams has been a superstar in full view for nearly two decades but still inhabits a complicated relationship with the public. I should come clean here: I think a lot of the griping about Serena Williams over the years has been utter nonsense. She’s had her unfortunate episodes—most memorably, a pair of verbal outbursts at the U.S. Open, which, while regrettable, hardly crack the upper echelon of Antisocial Moments in Tennis History. (Meanwhile, John McEnroe, who routinely menaced umpires, and Jimmy Connors, a habitual crotch-grabber, are celebrated as raffish antiheroes.) Only a fool would fail to see the obstacles and biases Williams has overcome, which she noted last December after winning Sports Illustrated’s Sportsperson of the Year.

One could make a compelling case that Williams is the greatest women’s player of all time—expert voices like Chris Evert now freely pronounce Williams the best ever. But the story of how she got here—with her sister Venus, African-American champions from Compton, California, schooled by an eccentric and prescient father, Richard—is so dazzling and improbable it’s almost fable-like. Today Serena Williams is at once a symbol of female and African-American power, a philanthropist, social media influencer, fashion designer and celebrity unlike anyone in women’s sports, who can stand shoulder to shoulder with any icon of her era. When Williams recently showed up in a video for Beyoncé’s new album, Lemonade—dancing alongside the beguiling pop queen in a Louisiana plantation home—her appearance felt both surprising and perfectly natural.

Williams did pick up a tennis racket, of course, and the rest is a history well-known and remarkable. In a few hours, Williams will get into a car for Miami, heading to the airport and Europe and a clay court season that will conclude in Paris at the 2016 French Open, where she is the defending women’s singles champion. At the end of June comes Wimbledon, where Williams is also the defending champion—she has won six times. Entering this year’s French Open, Williams had captured 21 women’s singles Grand Slam titles in her tennis career, putting her at the doorstep of Margaret Court (24) and Steffi Graf (22). She had also won 13 additional Slams in doubles and two more in mixed doubles, along with her four Olympic gold medals. At 34, she is the oldest women’s tennis player to ever be ranked No. 1.

To be a champion today requires an obsessive attention to detail. I ask Williams what she thinks TV announcers miss during tennis matches—what they don’t talk about but should. “I don’t think they talk enough about the lifestyle and what it’s [become] in the past 10 years,” she says. “Players used to have their parents. Or a coach. But now you wake up, sleep, eat…everything is tennis. The gym is tennis. Nutrition. Physio.”

In Williams’s living room, next to the clean white couch and a pod chair that hangs from the ceiling (“I saw it on Pinterest,” Williams says proudly), is perhaps the home’s most important piece of furniture: a massage table. At this point in her career, nagging aches and pains should be constant. But Williams says her body feels good when she wakes up. (Williams is an unapologetic night owl—it’s not uncommon for her to stay awake until 3 or 4 a.m. working on her various projects. “I try to skip the mornings.”)

Despite those extra practice serves, Williams did not win the 2015 U.S. Open, losing in a semifinal shocker to the unseeded Italian veteran Roberta Vinci, a defeat that was immediately categorized as one of the biggest sports upsets ever. Though Williams admitted the loss was painful and hard to put behind her, some perspective is useful. At that point, Williams had already won the 2014 Open and then the 2015 Australian Open, French Open and Wimbledon all in a row. The only thing denied was a fifth straight major and the symmetry of a “calendar” Grand Slam. There is not a tennis player in the world who wouldn’t take that kind of run. It was not exactly a disaster. Williams would move on. “There are very few champions on this planet, and they share things in common,” says Mouratoglou. “One of those things is the ability to forget the past…. They never look behind, always ahead.”

After the Open, Williams put down her racket for a while. She immediately threw herself into New York Fashion Week with a show for the Serena Williams Signature Statement line she sells on HSN, unveiling a line of fringe-inspired evening dresses and weekend wear. Vogue editor Anna Wintour and Williams’s then-rumored paramour, Drake, sat in the front row. After an upset loss, a lot of people would want to lie in bed and eat a tub of Häagen-Dazs. Instead there was Williams, walking down the runway, trading one pressurized centrifuge for another.

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Williams’s interest in fashion has long passed the point where it could be considered a part-time hobby. Both she and Venus (a designer herself) attended fashion school amid their tennis careers, and Serena obsesses over details like model casting and hair and makeup even as she plays in tournaments. Whereas a lot of celebrity collections are simply stars signing off on other people’s work, Williams considers it a point of pride that she knows “how garments work, fabrics work, how patterns are conceived.”

“Sometimes people just put their name on something and turn up at the event, and that’s it,” says Wintour. “Serena has the passion and the drive to make it her own, which is so different from many of these celebrities.”

Williams was once criticized for having interests outside of tennis. Both she and Venus heard it: They’re not focused. They’re not serious enough. Fashion school? You’ve got to be kidding me. Over time, however, it’s become apparent that those departures probably extended the careers of both sisters. Tennis is a game rife with early burnout, the byproduct of childhoods and adolescences devoted exclusively to one thing. Caring about something besides tennis isn’t bad. It’s healthy.

“I think it was great that they did that,” says Billie Jean King, the women’s tennis legend who played singles until she was nearly 40 and doubles even longer. “Why not? People should have the journey that they want.”

Today, Williams’s off-court passions are part of what fans love about her; it’s where they get to see the other Serena, who’s more relaxed and a bit of a goof. “She’s really funny,” says her close friend and tennis competitor Caroline Wozniacki. “We both think we’re hilarious and could be comedians.”

By now everyone in tennis knows that Serena Williams is a karaoke warrior (“That’s her thing,” says Wozniacki), but she also continues to be fascinated with dance. Each year, she and Venus host a competition called the Williams Invitational in which teams compete in, among other events, dance routines. Then there was Williams’s appearance in Beyoncé’s “Sorry” video, which Serena had to keep confidential for months.

“If you tell me, ‘Don’t say anything,’ I won’t say anything,” Williams says.

“She really kept it a secret,” says Wozniacki, who didn’t know about the video until she saw it.

“Sorry” is an edgy hit. But Williams’s favorite dance song is more of a vintage choice: “Conga,” by Gloria Estefan and Miami Sound Machine. Her favorite TV show? The Golden Girls. (Yes: I will point out that both “Conga” and The Golden Girls debuted in 1985, when Williams was a toddler.) She has never watched Game of Thrones (“I saw the first five minutes, and I was like, ‘This is waaaaay too much for me’ ”). She wishes she got out more. “I am too much of a hermit,” she admits. “I go through phases—I was going out to dinner a lot, but I’ve [reverted] to my old, staying home thing.”

If Serena Williams could spend a day following one person, who would it be? “Elon Musk,” she says. “I’d follow him to see what he’s doing with all his innovations, all of the stuff he’s creating. I’d love to figure out where he’s going.”

WILLIAMS’S 2016 did not begin like her steamroller 2015. In January, she lost the Australian Open final in three sets to Germany’s Angelique Kerber, then in the Indian Wells final to Victoria Azarenka. She finally won her first tournament in mid-May at the Italian Open, a few days after a bizarre incident in which she snacked on some of Chip’s hotel room-service dog food and got an upset stomach (an episode discussed in detail on Snapchat).

“Not as great as I want it to be,” says Williams, assessing her year so far. “I could do better. But honestly, that’s how I felt about 2015.”

Just to clarify: In 2015, Serena Williams went 53–3.

When I ask what she’s most proud of in her career, she doesn’t mention a medal or a Slam. She says she’s most proud of her return to Indian Wells, the California tournament she and Venus shunned for nearly a decade and a half after Serena was mercilessly booed in the 2001 final, a couple of days after Venus had withdrawn from their semifinal match with an injury. “I don’t blame them for not playing there,” says Billie Jean King. “People treated them horribly. I was there. I think it’s great she went back.”

“To be forgiving,” Williams says. “I am proud of that.”

Oddly, Williams found herself drawn into another Indian Wells controversy at this year’s tournament, when the event’s CEO, Raymond Moore, made a series of tin-eared remarks on the topic of equal pay in the sport, claiming, among other things, that if he were a “lady player” he would “go down every night on my knees and thank God Roger Federer and Rafa Nadal were born, because they have carried this sport.”

“We, as women, have come a long way,” Williams said at a tournament press conference shortly before Moore would resign under pressure. “We shouldn’t have to drop to our knees at any point.”

Even as an ensconced World No. 1, Williams cannot avoid questions about her tennis mortality. Mouratoglou says that at this point, Williams is in it for the Slams—“I do not think that her ranking is important to her”—and thinks she will stop only when she does not believe she can win major tournaments. Williams does not have a reliable nemesis on tour; the women’s game’s second-most-decorated player, Maria Sharapova, is currently serving a suspension for taking a banned prescription drug, but Sharapova is hardly a rival—she hasn’t beaten Williams in more than 10 years.

King thinks Williams has plenty left. “There are lots of records to break, if she wants it,” she says. “When you get to her age, you have to decide how much you’re willing to invest physically, emotionally, if the return on investment is worth the effort. And the effort is more as you get older, not less.”

“I think it will hit me,” Williams says. “I’ll just have the feeling of ‘I don’t want to do it anymore.’ ”

The end does not appear to be imminent. As for Williams’s post-tennis life, there’s her fashion business, which she hopes to expand. There are her philanthropic efforts, like the schools she sponsors in Kenya and Jamaica. When I ask Williams if she could see herself in a TV booth during a tennis match, she won’t rule it out. (The only idea she gives a stern no to is coaching.)

“She has so many avenues open to her I can’t imagine her being defined by one choice,” says Wintour. “You give up your life to go on that circuit. I imagine she’ll enjoy the freedom after tennis.”

“Being a mom would be fun,” Williams says. How many children?

“Twelve,” she says, deadpan. “A baker’s dozen,” she continues, laughing.

Williams’s life remains such a whirlwind that she fantasizes mostly about taking a break. A perfect day off, she says, would be sitting at home on the couch watching Netflix and Investigation Discovery, and maybe taking a dance class. But such days almost never happen. Those ordinary things like amusement parks, grocery stores and trips to the mall that Williams sometimes craves still feel far away. They might not happen for a long while, because Serena Williams picked up that tennis racket and built an extraordinary life.

Write to Jason Gay at Jason.Gay@wsj.com