As the shutdown of the U.S. government enters its third day, disrupting everyone from farmers who can't cash their paycheques to Statue of Liberty tourists, a growing cohort of Republican politicians is openly taking party leadership to task for a tactic that they fear will incur a deep backlash from the American public.

"It's crazy. I don't understand the whole point, the whole strategy. Most Americans don't understand it," Congressman Devin Nunes, a Republican member of the House of Representatives from California, told CNN late Wednesday night.

"The Democrats are giddy about this behind closed doors…. This is benefiting them politically."

"The question is, does the shutdown advance our goals? I hold the view that it doesn't, and I've said so," Representative Scott Rigell, a Virginia Republic, told the Fox News channel.

- The blame game: Republicans, Democrats and the shutdown

- Graphic:What's open and what's closed?

- Analysis: Neil Macdonald on the perverse math that enables the Tea Party 

House Republicans adamant about cancelling, delaying or watering down President Barack Obama's signature health-care reform legislation have refused to pass spending measures without anti-Obamacare provisions.

Senate Democrats insist that the Affordable Care Act, as it's formally known, was passed into law long ago and isn't up for reconsideration.

With 800,000 federal government employees forced into taking leave, some agencies have almost entirely shuttered, including NASA, the Commerce Department and the Environmental Protection Agency.

The Republican-controlled House has tried to push piecemeal legislation to fund individual departments and programs, but most Democrats, including Obama, want a comprehensive resolution that would reopen the whole government, not just bits of it.

Obama said Thursday morning that a majority of House Representatives actually want to pass a comprehensive funding bill and that "the shutdown could end today," but "extremists" within the Republican party are preventing a simple yes-no vote. 

"They won't agree to end the shutdown until they get their way," he said in a speech in Maryland.

'Silly strategy'

Meetings Wednesday night between Obama and congressional leaders failed to resolve the impasse, and a small but growing group of House Republicans are now openly criticizing the rightist, Tea Party-led faction within their party that has been demanding clawbacks to Obamacare as a condition of reopening the funding taps.

"It was a silly strategy from the beginning," Nunes said on CNN. "I just don't think as long as there's a guy named Obama in the White House that you're going to get rid of Obamacare without a veto."

The California congressman said his more radical caucus colleagues should have known that any legislated effort to roll back the Affordable Care Act would almost certainly be vetoed by the president — which can be overturned only by a two-thirds vote of both the U.S. House and  Senate.

"I thought  from the beginning that that was, really, couldn't be done, and it's really just a matter of math."

Where a half-dozen House Republicans originally dissented publicly from the party's stance, the number has grown. Some moderate Republicans now say they think around 25 of their colleagues would openly support a straight-up bill to reinstate government spending.

Combined with all 200 Democratic votes in the House, that could be enough to pass such a bill — but only if it could surmount the considerable procedural hurdles in bringing it to the floor against the majority Republican leadership.

Small hope for compromise

Another small hope being discussed on Capitol Hill is that Democrats could concede on eliminating a minor piece of Obamacare, a 2.3 per cent tax on medical devices that even some Democratic senators oppose. The thinking is that could be enough to allow Republicans to save face and agree to a compromise.

"The challenge for us, and the solution for us, is to find the middle ground to end the impasse," Republican Representative Erik Paulsen told Minnesota Public Radio. "The medical device tax, I think, is the one vote that had the most bipartisan support as an example."

- U.S. government shutdown could make Washington a smelly city

- U.S. government shutdown scuppers weddings, KKK rally

The shutdown stalemate is already rattling investors. Stock markets in the United States faded for the second day in a row on Thursday, and Europe's top central banker, Mario Draghi, called the shutdown "a risk if protracted."

As the politicians battled, mail continued to be delivered, air traffic controllers remained at work and payments were being made to recipients of Social Security and unemployment benefits.

As well as the closure of national parks, halted were:

- Most routine food inspections by the U.S. Food and Drug Administration.

- Census and economic data processing by the Commerce Department.

- The work of automobile safety experts.

- A wide array of government research. 

Related on HuffPost:

Loading Slideshow...
  • The Deficit Has Grown Mostly Because Of The Recession

    The deficit has ballooned not because of specific spending measures, but <a href="[1][id]=FYFSD" target="_hplink">because of the recession</a>. <a href="" target="_hplink">The deficit more than doubled</a> between 2008 and 2009, as the economy was in free fall, since laid-off workers paid less in taxes and needed more benefits. The deficit then shrank in 2010 and 2011.

  • The Stimulus Cost Much Less Than Bush's Wars, Tax Cuts

    Republicans frequently have blamed <a href="" target="_hplink">the $787 billion stimulus</a> for the national debt, but, when all government spending is taken into account, the stimulus frankly wasn't that big. In contrast, <a href="" target="_hplink">the U.S. will have spent nearly $4 trillion</a> on wars in the Middle East by the time those conflicts end, according to a recent report by Brown University. <a href="" target="_hplink">The Bush tax cuts have cost nearly $1.3 trillion</a> over 10 years.

  • The Deficit Grew Under George W. Bush

    When George W. Bush took office, <a href="" target="_hplink">the federal government was running a surplus</a> of $86 billion. When he left, that had turned into a $642 billion deficit.

  • The Deficit Is Shrinking

    <a href="" target="_hplink">Last year's federal budget deficit</a> was 12 percent lower than in 2009, according to the Office of Management and Budget.<a href="" target="_hplink">The deficit is projected to shrink</a> even more over the next several years.

  • Investors Are Paying Us To Borrow Money

    <a href="" target="_hplink">The interest rate on 10-year Treasury bonds</a> is <em>negative</em>, according to the Treasury Department. Investors are even paying us for 30-year Treasury bonds, when adjusted for inflation.

  • Investors Are Not Running Away

    <a href="" target="_hplink">Conservative commentators</a> have been warning for years that investors will run away from Treasury bonds because of the national debt. So far it's not happening. <a href="" target="_hplink">Interest rates on Treasury bonds</a> continue to hover at historic lows.

  • Health Care Reform Reduces The Deficit

    <a href="" target="_hplink">Republicans have blasted the Affordable Care Act</a> as "budget-busting." But <a href="" target="_hplink">health care reform actually reduces the deficit</a>, according to the Congressional Budget Office.

  • The U.S. Is Borrowing Less From China

    <a href="" target="_hplink">The U.S. government is borrowing much less from foreign countries</a> than before the recession, according to government data cited by Paul Krugman. That is because the U.S. private sector is financing our bigger deficits.

  • We Spend A Lot On Defense

    <a href="" target="_hplink">Defense spending constituted 20 percent</a> of federal spending last year, or $718 billion, according to the Center on Budget and Policy Priorities. This adds up to <a href="" target="_hplink">41 percent of the world's defense spending</a>, according to Bloomberg TV anchor Adam Johnson. <a href="" target="_hplink">Mitt Romney has vowed</a> to not cut defense spending if elected president.

  • We Spend A Lot On Health Care

    <a href="" target="_hplink">Health insurance, including Medicare and Medicaid, constituted 21 percent</a> of federal spending last year. In contrast, education constituted 2 percent of federal spending. Meanwhile, <a href="" target="_hplink">Mitt Romney and Paul Ryan have promised not to change Medicare</a> for Americans age 55 and older.

  • Republicans May Want Large Deficits For Now

    <a href="" target="_hplink">The federal budget deficit ballooned</a> under Ronald Reagan, and that may be just the way Republicans like it. <a href="" target="_hplink">Some Republican thinkers</a> have proposed <a href="" target="_hplink">"starving the beast"</a>: that is, cutting taxes in order to use larger deficits to justify spending cuts later. Since Republicans ultimately want lower taxes and a smaller government, what better way is there to cut spending than to make it look urgent and necessary?