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Saint Ann’s, an elite private school in Brooklyn Heights, charges up to a whopping $42,000 a year in tuition, counts Lena Dunham, Jennifer Connelly, and Jean-Michel Basquiat among its esteemed graduates, and is so sought after that it wasn’t even willing to squeeze Matt Damon’s three kids into packed classes this school year.

Yet, according to Uma Thurman’s ex, French financier Arpad “Arki” Busson, the institution is a dump, with students in need of a makeover.

During an ongoing custody trial, Busson’s lawyer Peter Bronstein said the French financier was at odds with Thurman over sending their daughter, Luna, 4, to Saint Ann’s.

“He felt they had small classrooms, they were cluttered, that the atmosphere was depressing, that the kids looked dirty and disheveled,” Bronstein declared in court last week, adding that it “doesn’t compare with Chapin,” a buttoned-up all-girls private school on the Upper East Side.

“It was probably my kids that he saw,” one Saint Ann’s mom said, laughing, after reading the quotes in Page Six.

At the exclusive Saint Ann’s, being a little messy is part of the appeal.

“Honestly, there is a grunginess to the facility,” said a Saint Ann’s mother who lives in downtown Manhattan and sends her son to the high school.

“When I visited the school it had less polish and shine than the other schools we visited,” she admitted. “Our tour guide, who was a student, sensed my hesitance. He said, ‘We love it here and don’t want it to change much. It has a soulfulness.’ ” (She adds that the main school is currently undergoing renovation.)

Saint Ann’s has been challenging the status quo since it was established in 1965 and led by founding headmaster Stanley Bosworth, an eccentric scholastic who bragged that he wanted Saint Ann’s to produce 10 percent of the nation’s poets, according to a 2004 New York magazine profile. (Bosworth died in 2011.) Part of his shtick upon meeting applicants’ parents would be to dramatically proclaim, “Teenagers are sexual!” and excitedly await their reactions.

The school, a self-professed “amusement park,” was first housed in the basement of Saint Ann’s Episcopal Church. Now grades 4-12 are housed in the 13-story main school close to the Brooklyn Heights promenade, with lower grades down the block. The preschool sits nearby in a former church-turned-community center and the kindergarten is a block away.

And while the Saint Ann’s teen smoking lounge — which elected a king and queen each year according to the New York magazine article — has been abolished for decades, the institution still gives its students more freedom than your typical NYC private academy.

There’s a funny urban myth [about the school] that if you don’t want to take your math test, you can do an interpretive dance instead.

Over the years, the school has offered classes ranging from puppetry to kite making. Juniors and seniors have the luxury of designing their curricula. Preschoolers act in operas and films about ancient Greece.

“My 12-year-old daughter’s friend who goes to Saint Ann’s was groaning about her art class because she had to paint water without sand and without sky, just looking from above,” said a Park Slope mother whose daughters attend another Brooklyn private school.

Saint Ann’s famously doesn’t have grades; teachers instead pen full-page reports for each student. One former student said, “It was a mix of positive reinforcement and encouragement in addition to being honest about the student’s struggles.”

“How do you give a grade on an oboe’s sweet, beautiful sound?” Bosworth mused to New York magazine.

“There’s a funny urban myth [about the school] that if you don’t want to take your math test, you can do an interpretive dance instead,” the Park Slope mom said. “You can freestyle your education.”

The downtown Manhattan mother said Saint Ann’s no-grade policy helps kids “be really intrinsically motivated.”

“I have found, looking at some of my son’s peers, that they are afraid to try things that they may not do well at because it may mess up their college record. It may sound pretentious, but Saint Ann’s is going to attract people who are interested in education and not just what top college they can get into.”

Still, the students tend to get into some pretty great schools.

Saint Ann’s is consistently ranked one of the country’s top 50 high schools. For the 85 graduates in the 2016 class, there were 26 offers from Ivy League schools.

Not that Saint Ann’s cares about such bourgeois merits.

“When I was there, Stanley told me not to go to college and go be an artist because he really liked my paintings,” 31-year-old Jamie Singer told The Post. She attended Saint Ann’s alongside actresses Paz de la Huerta (“Boardwalk Empire”) and Jemima Kirke (“Girls”), and now runs Ussie, a visual messaging app. She ended up enrolling at Cornell.

“That’s a really unique situation, to have a headmaster come in and tell a student not to go to college.”

Singer said it was only once she went to university that she truly grasped how different Saint Ann’s was from other academic institutions. “The word ‘hipster’ didn’t exist yet, [but] I finally realize that Saint Ann’s is where that all started.”

Dana Betts, 37, was a student at Saint Ann’s from first grade through 12th, and is now a jewelry designer and professional squash player. She also coaches the squash team at Nightingale-Bamford, an all-girl Upper East Side private school.

“Saint Ann’s just seemed so much more free and diverse [than Nightingale]. People could be anything they wanted,” said Betts.

Betts says that unlike schools such as Chapin, where showing your status through a Van Cleef necklace and pearl earrings is the norm, at Saint Ann’s “there are a lot of people who come from money, but they dress like they don’t.”

At Saint Ann’s, individuality was the biggest status symbol of all, Singer explained. “There would be kids who come to class in tutus but it was almost unacceptable to have a Dior bag in school,” she said.

Getting into Saint Ann’s can be harder than getting into a top Ivy. With only 79 to 85 students per grade in the middle and high schools (and fewer in the younger grades), there are limited spaces, as Matt Damon found out last summer when he tried to enroll his three daughters and was rebuffed.

Unlike posh Manhattan institutions where financiers’ families rule the roost, “Stanley wasn’t looking for your average Wall Street family,” said Singer. “If you had a lot of artists in your family, that was specifically something that Stanley sought out.”

Basquiat, who attended Saint Ann’s at age 7, received a scholarship thanks to his artwork.

Applicants take an IQ test and go through an interview process. The school’s Web site states that they are looking for students “who show evidence of the intellectual motivation and aptitude that will enable them to handle both freedom and rigorous academic requirements.”

“Not only did you have to be smart, but you had to show that you had something to offer the intellectual community,” said Saint Ann’s alum Singer.

And alumni shouldn’t assume that their own kids are guaranteed to get in just because of family ties, either.

“I have friends from Saint Ann’s who thought that ‘of course, my kids will automatically get in,’ and they’re told, ‘Sorry, there’s no space. We will put you on the waitlist,’ ” said Betts.

But once a kid’s in, it’s up to the student to carve his or her own destiny.

“It’s a place where it’s OK to be dirty, it’s OK to play. With a lot of [other private schools], you don’t always get that permission,” said Singer.

“We’ve gotten to study at the Hogwarts of New York,” she added, referring to the magic school atended by Harry Potter. “I feel blessed.”