Even 4,000 miles from Washington, and four years removed from her concession speech in the 2008 Democratic primary race, where she thanked supporters for putting “18 million cracks” in the highest glass ceiling, Hillary Rodham Clinton still gets the question.

If anything, since making it clear that she wouldn’t serve in a second Obama administration, she has started getting it more. In Copenhagen last month, after meeting with top Danish officials and taking reporters’ queries on U.S. policy toward Syria, the secretary of state was pressed on her political ambitions.

She did not sound like someone contemplating another campaign. “I’m looking forward to working as hard as I can until the end of my tenure as secretary of state, and then will look forward to some time to collect myself and spend it doing just ordinary things that I very much am looking forward to again,” Clinton said, “like taking a walk without a lot of company — not that I don’t love seeing you all — but just having the time to set my own schedule and pursue a lot of the interests that I have pursued my entire life, particularly on behalf of women and children.”

The questioner persisted. “No politics?”

“No politics,” Clinton replied.

She has offered some version of that response for years now, always suggesting that her service as America’s top diplomat will be her final official role in public life. But that hasn’t stopped the speculation about what she should or shouldn’t do.

There was the chatter earlier in President Obama’s term that she might consider a primary challenge against the president for the 2012 nomination. (An absurd notion, her advisers said.) Or, more recently, that she would swap roles with Vice President Biden. (Even more absurd, advisers to Biden, Clinton and Obama declared.) Or that she might take over the World Bank or the Pentagon, or join the Supreme Court — none of which has come to pass.

But what of 2016? Has Clinton given up on busting though that glass ceiling, or will her ambition and influence move her to run for president once again? In recent days, prominent Democrats such as former House speaker Nancy Pelosi, former Pennsylvania governor Ed Rendell and her husband, former president Bill Clinton, have speculated aloud that she just might.

Guessing about Hillary Clinton’s presidential ambitions has been a game since at least 2003 — when she began laying the groundwork for a run — and has never stopped. Back then, her friends admit, she had to be coy. Now, they say, Clinton truly does not think she will run again.

The speculation today stems from her soaring popularity — in a Washington Post-ABC News poll in April, her favorability rating stood at 65 percent, her highest ever — as well as from her success as a Cabinet official and from the residual affection she earned during her presidential campaign, not to mention what some Clinton advisers call the “buyer’s remorse” of Democrats who wonder whether she would have been a better president.

Philippe Reines, a longtime Clinton aide who is now deputy assistant secretary of state, described the chatter as mere “cable catnip.”

“I can already hear Wolf Blitzer’s voice, 25 years from now, intoning: ‘Happening again, for the ninth cycle in a row, speculation is reaching a fever pitch about a possible Hillary Clinton run in 2040 — her approval rating is an eye-popping 137 percent, and she’ll only be 93 on Election Day,” Reines said.

Whether she runs or not, Clinton is almost certain to spend 2013, at least, unshackled from the demands of government service and entering an uncharted phase in her life, as she follows her husband into the private sector and, most likely, the nonprofit world.

She is strongly considering joining her husband’s philanthropic outfit in some fashion, setting up her own office at either the Clinton Foundation, which he formed in 2001, or its subsidiary, the Clinton Global Initiative, to continue her work on women’s issues and human rights. The Clintons would form a formidable fundraising team and could do more effective work together than if they entered an awkward competition at separate nonprofits, their allies say.

Still, it would mean that for the first time since 2001, when she went from first lady to senator from New York — later becoming the most successful female presidential candidate in U.S. history — Hillary Clinton would be linking her professional life to her husband’s.

Though Clinton will also quickly become one of the most sought-after and well-compensated speakers on the planet, it is unclear how much time she wants to devote to the speech circuit or to political fundraising, according to friends. If anything, they say, she has enjoyed being able to turn down requests over the past three years, as required in her role as secretary of state.

Instead, Clinton is expected to write a book — either a sequel to “Living History,” her best-selling memoir from 2003, or possibly something more policy-centered. She can command a multimillion-dollar advance, perhaps doubling the reported $8.1 million advance she got in 2003, when her book sold more than 1 million copies in the first six weeks. (And yes, money could be a consideration for the once-struggling Clintons, despite their more recent financial stability.)

Of more than half a dozen Clinton friends and advisers, past and present, who talked candidly about her prospects, most did so on the condition that they would not be quoted. They were split: Some thought she should not run again, while others expressed strong interest in her doing so. But all agreed that she has fully rehabilitated herself since 2008.

Indeed, Clinton has appeared even more at ease as secretary of state than she did as a senator, helped along by a wonkish press corps more focused on the details of multilateral negotiations than on what she is wearing. She has sidestepped diplomatic land mines, earning credit as a leader in winning situations — such as the Libya uprising that, with U.S. help, toppled Moammar Gaddafi — without being bogged down by losing ones, such as the protracted Israeli-Palestinian conflict or the dwindling U.S. war in Afghanistan.

It is precisely Clinton’s current ease — evident, for instance, in her playful participation in the “Texts From Hillary” Internet spoof — that leads some of her more protective confidants to believe that she will not, and should not, seek the presidency again.

In the past four years, Clinton recovered from a devastating defeat and made an improbable leap to become a trusted adviser to her formal rival. If she were to run and lose again, a similar recovery would be much more difficult, and the twin defeats could eclipse her life’s work.

The last campaign took its toll. She weathered pronounced sexism. Her team was constantly at war with itself. Above all, Clinton had to take hard stands and try to inspire voters, all while subject to around-the-clock scrutiny. One cautious friend suggested that if Clinton were to run again, her popularity would drop overnight because she would once again be center stage, a position that makes all politicians more polarizing.

Then there is the other camp: Those who desperately want her to run, fulfilling the hopes of 2008 and the aspirations of those who saw her as the first female president. She has proved herself as a global leader, they argue, and is the most qualified candidate to follow Obama. The chance to make history could outweigh her reluctance after an adequate break.

Maybe the campaign wouldn’t be so hard next time, they argue. Clinton has less to prove now. She wouldn’t have to convince anyone that she is tough enough to be commander in chief. To the extent that ideological categories apply, she has been to the right of Obama and Biden on challenges from Libya to Afghanistan. For a woman aspiring to higher office, she has checked what is arguably the most important box.

And she now reflects disparate aspirations: Among some liberals, Clinton is seen as a potentially better standard-bearer than Obama, whose compromises on health care and the budget have frustrated his party’s base. To foreign policy hawks, she is pragmatic and unafraid of U.S. dominance. To Washington insiders, she is a like-minded realist who would play hardball with Congress and understands how the system works.

Of course, numerous other candidates are already lining up for 2016, their ambitions premised in part on the belief that the Clinton moment has passed. But in the era of eternal campaigning, Clinton has something her possible rivals do not: time.

Potential 2016 Democratic hopefuls such as Maryland Gov. Martin O’Malley and New York Gov. Andrew Cuomo will need to start laying the groundwork for a national campaign right away. So will the women who could run, such as Homeland Security Secretary Janet Napolitano or Sen. Kirsten Gillibrand (N.Y.). The only other candidate who could take his time may be Biden, who would run from a White House perch — but would not be formidable enough to clear the field.

Clinton, because of her political network and experience, could decide as late as 2015, although that would surely frustrate donors and the other contenders.

“It’s partly amusing, partly annoying, that everyone is sitting here trying to map out the best political move for her,” said Mo Elleithee, who worked for Clinton in 2008. “If she wants to get back into the game, she has time to make that decision. . . . Everyone else is going to start gearing up the day after this election is over.”

The outcome on Nov. 6 could play a role in how events around Clinton unfold. If Obama wins reelection, she would be unlikely to start staking out independent views on policy issues. But if Mitt Romney wins, she could be motivated — by competitiveness, even anger — and would almost certainly face pressure to mount an opposition bid.

And if she did decide to run? “It would look very different,” one friend said.

One key change would be her advisers.

At the State Department, her operations are largely managed by longtime confidante Cheryl Mills. She has jettisoned some of her former political advisers and consultants, drawing closer to her foreign policy advisers and forming an inner circle that is a mixture of old aides and new ones: State Department Director of Policy Planning Jake Sullivan, Deputy Secretary of State Tom Nides, former assistant secretary for legislative affairs Rich Verma, Center for American Progress founder John Podesta, and longtime aides Capricia Marshall, Reines, Huma Abedin and Lona Valmoro.

Unlike four years ago, Clinton is not getting constant political briefings from strategists such as Mark Penn (although Penn, who wasn’t banished from her circle, is still in occasional touch).

“Her inner circle has changed,” one friend said.

Another person still in her orbit also said the cast would be different in 2016 than it was in 2008. “It’s not like she doesn’t know she lost. Do people really think she would run the exact same campaign again, have the exact same people who screwed it up?”

Anne E. Kornblut is a deputy political editor at The Washington Post and the author of “Notes From the Cracked Ceiling: Hillary Clinton, Sarah Palin, and What It Will Take for a Woman to Win.”

Read more from Outlook, friend us on Facebook, and follow us on Twitter.