On Saturday night, a Supreme Court justice, two senators and the chief executives of McDonald’s and Lockheed Martin will officially join a secretive private club that meets just once a year, has no official purpose, and is named after a plant that will do almost anything for a drink.

And despite that — or perhaps because of it — the Alfalfa Club is one of the most prestigious organizations in Washington.

The 101-year-old club boasts a unique mix of politicians, administration officials, rainmakers, top military brass and corporate leaders from all over the country: Billionaires Warren Buffett, David Rubenstein, Michael Bloomberg, Bill Marriott and Steve Case; Chief Justice John Roberts, White House senior adviser Valerie Jarrett, George H.W. Bush and George W. Bush, Dick Cheney, Sandra Day O’Connor, Colin Powell, Madeleine Albright and Vernon Jordan, to name just a few.

It’s a cross-section of power brokers so influential that almost every president has made a pilgrimage to the annual gathering. President Obama has addressed the group twice: a week after his inauguration in 2009, and again in 2012 when he joked about escaping the White House bubble: “One of my big goals this year was to get out and be among everyday, ordinary Americans — like the men and women of the Alfalfa Club.” Big laugh.

Why, you ask, would men and women with all the money and fame they could ever use care about Alfalfa? Because they’re never too rich or too successful to pass up an evening that really is like none other, where so many big shots from different spheres bond together over cocktails, jokes, lobster and filet of beef.

“The members are certainly some of the most respected people in America,” said philanthropist Adrienne Arsht, who’s attending again this year as a guest of O’Connor. “It’s not about business, it’s about relationships . . . which is about business.”

“I’ve never seen a dinner like it anywhere else,” said Raul Fernandez, tech entrepreneur and vice chairman of Monumental Sports and Entertainment, the company that owns the Washington Wizards and Capitals. Fernandez will be inducted at the 2014 dinner Saturday in the crop of new members — affectionately referred to as “sprouts” — that includes Supreme Court Justice Stephen Breyer, Sens. Timothy M. Kaine and Kelly Ayotte, Lockheed Martin chief executive Marillyn Hewson, McDonald’s chief executive Donald Thompson, Aspen Institute president Walter Isaacson, former head of General Dynamics Jay Johnson, James A. Baker IV and financiers C. Bowdoin Train and William Grayson.

“Mark Warner has always told me how fun it is and I was honored to be asked to join,” said Kaine.

The evening, always closed to the press, is described by members as a reunion, a party, a chance to enjoy a few laughs and silly speeches. “It is an occasion for old friends to get together and where new friends can be made,” said incoming president Sen. Dianne Feinstein (D-Calif.). “The evening is filled with humor — some good and some not so good.” The 200 members are allowed to bring two guests each to the dinner held at the Capital Hilton; the seating is intimate, picnic-style with everyone smooshed together at long, narrow tables. Between speeches by the outgoing president — this year it’s Sen. Lamar Alexander (R-Tenn.) — the incoming president, and the club’s annual mock nominee for president (always a surprise), the aisles are “absolutely clogged,” reported one guest. “It’s like a seething anthill. What are they doing? Pressing the flesh.”

It’s easy to dismiss Alfalfa as a bastion of the privileged, a haven for America’s ruling class to celebrate itself. Occupy D.C. staged a loud protest outside the 2012 dinner, glitter-bombing the politicians and one-percenters who dared cross the hotel threshold. But that didn’t prevent the dinner guests from showing up, primarily because many rejected the narrative of themselves as disengaged elites.

“I don’t see it that way,” said businesswoman Catherine Reynolds, who’s bringing Virginia Gov. Terry McAuliffe (D) and first lady Dorothy McAuliffe as her guests this year. “I see it as a group of people who are doers in their industry and in their community. They try to get things done.” The spirit of the night, she described, is one of “camaraderie and patriotism. I think that’s what makes it all jell.”

Talk to a few members and you hear versions of the American dream realized. Much of the appeal of Alfalfa is the sense that accomplishment, not class or money, is a major factor in getting in. People who started with nothing and made it big, Alfalfans who are proud and still a little amazed at their good fortune. “Bloomberg and I always joke, ‘How did two blue-collar kids from Medford, Massachusetts, end up at the Alfalfa Club?’ ” said longtime member Bill McSweeny, philanthropist and former president of Occidental. “There’s a lot of that.”

The club started in 1913, when — as the story goes — four friends gathered to celebrate the Jan. 19 birthday of Confederate general Robert E. Lee. They took their name from the alfalfa plant, known for roots that travel deep to find refreshment. The annual dinner, which quickly grew in numbers and prestige, is always held on the last Saturday in January. Richard Pearson, a member for more than 50 years, can’t remember the dinner ever being canceled, even during snowstorms and power outages. “I’ve been to somewhere we ate in our overcoats,” he said.

It’s easy to see why politicians flocked to Alfalfa. Prescott Bush, former senator and the late patriarch of the Bush clan (the Alfalfa’s devoted, unofficial first family — George W. Bush attended all eight years he was president) “thought the club was very important because it’s a power center of politics and money,” said writer Landon Parvin, who was inducted in 2002 and is responsible for many of the wittiest lines in Alfalfa speeches.

Alfalfans embraced tradition and establishment and resisted change. African Americans were not admitted until 1974. In 1986, Sen. Paul Simon drilled Supreme Court Justice William Rehnquist about Alfalfa during his confirmation hearing for chief justice, suggesting the men-only rule might violate American Bar Association rules prohibiting membership in discriminatory private clubs. Rehnquist replied that the club meets annually to “hear some patriotic music, listen to some funny speeches, and go their way for the rest of the year.”

“Sounds like part of the old-boy network to me,” sniffed Simon, who asked the justice to consider quitting. (He didn’t.) Nine years earlier, legendary Washington lawyer Clark Clifford had urged then-new HHS Secretary Joe Califano to join the club. “You’ll be returning to law practice here,” Califano wrote in his 2004 memoir. “It can help you.” Califano, already under fire from women for his opposition to federal funding for abortion, declined. Clifford dismissed his concerns and warned that there wouldn’t be a second offer. “No one turns down the Alfalfa Club.”

It was just 20 years ago that women were finally allowed to join after a strong push from another HHS secretary, Donna Shalala, and a snub from President Bill Clinton. O’Connor, Elizabeth Dole and Katharine Graham were the first women inducted. O’Connor went on to become the club’s first female president and its only presidential nominee; Feinstein will be the second. “Perhaps it will be a trend, but I doubt it,” said the senator.

Alfalfa is especially valuable for women leaders, Arsht said. “The women in the room illustrate, almost in a subliminal way, women of accomplishment. Gravitas is a given.” Or put it this way: Instead of a female chief executive or senator being mistaken for someone’s wife (yes, it still happens), a spouse is more likely assumed to be a corporate heavyweight.

But good luck getting in. The membership process is “very secretive and murky,” joked Parvin. “The anointed just appear.” Well, not quite: Alfalfa tries to keep membership to about 200; openings only occur when members die. There are no formal letters of recommendation, said Pearson, the longtime club secretary, but prospective members are quietly approached after they’ve been proposed and approved by the “powers that be” (some of the older members). Lobbying for admission is frowned upon. The club tries to recruit an equal number of Democrats and Republicans, although the corporate types skew slightly conservative. Oh, and no meanies allowed. “For the most, they have to have good personalities,” Pearson said.

“It’s an honor to be invited,” said Fernandez — who, at 47, is a decade or two younger than most of his fellow Alfalfans. “It’s a live connection to a different era that lives on.”

Or, as Parvin put it: “You can be any age and be an old fart. We’re trying to uphold that tradition.”