The King of Belgium played a significant role in the 2007 election aftermath mess.
I was rather amused at the time that the Prime Minister attempted to resign and the King told him he couldn’t. I pictured old Albert telling Leterme: “No, you can’t resign! Now get back in there, you coward!” 
In Norway the royals have been known to make very general semi-political statements, along the lines of “This particular issue is of particular concern to me.” To say that they approve or disapprove of any particular policy is however taboo. The government (in the parliamentary sense of the PM and his or her cabinet) meets with the King once a week, but this mostly consists of telling the King what they’ve done in the previous week and what they intend to do in the near future. At most they might ask for the King’s opinion, which they may or may not ignore.
The King has to sign every piece of legislation into law, and technically he can choose not to do so thereby vetoing it. In practice he doesn’t dare. The King also invites one party or a coalition of parties to form a government after each election and in the event that a government falls between elections, and technically he is free to choose whichever party or coalition he chooses. In practice he must choose the one that won the election or is most likely to be able to govern. This was established beyond a doubt back in 1928 when the Norwegian Labor Party - at the time a good bit further to the left than it is now - won an election and were the obvious candidates to form a government. King Haakon’s advisors and friends were shocked and said that surely he wouldn’t allow those communists to take over the government?! To which he answered: “I am also the communists’ king.” (That government lasted two weeks, but the precedent was forever.)
I need to construct an analogy to begin what I have to say here: Did you know that in the U.K., the highest judge in the country is also a member of the Prime Minister’s Cabinet and the presiding officer of the House of Lords? It’s true – he’s the Lord Chancellor.
Odd mix of duties, right? Well, for us Americans, that’s how parliamentary democracies look at the idea that our President is simultaneously Head of State, the sovereignty of the nation embodied in one man, and Head of Government, the elected political leader nominated by one party and elected by (indirect) popular vote.
In the U.S., the President cannot enact a law, or cause a law to be enacted; that’s Congress’s job. He can propose ideas for good law, he can bring a lot of influence to bear – but if he cannot get 218 Congressmen and 51 Senators to agree with it, he’s beat, rabbit!
In the parliamentary democracies, the Premier/Prime Minister is the man who leads a majority in the strongest house in the legislature – the House of Commons in British and Canadian parlance. If he decides it’s worth putting his prestige on the line to get a measure passed, he can guarantee it will get a majority – of course, he risks his own office in the process, but he has the power to push it through.
Now, what keeps this from being a dictatorship is two things: in theory, his supporters in the legislative house can reject one of those lay-it-on-the-line measures, in effect voting him out of office; and the whole lot of them --legislators, P.M, Cabinet, and all – must stand for re-election at least once every five years. Or, actually, three: it’s the monarch (or in parliamentary republics, the President) who names the P.M. to his job. The monarch has a restricted choice in whom he will name: it has to be someone who can hold a majority in the legislature. If, say, the Conservatives hold a majority of seats in the legislature, then the monarch’s choice is down to one: the leader of the Conservative Party.
The other half of this picture is the Head of State – the person who speaks for the nation, who embodies the national sovereignty, who acts ceremonially with common assent to formalize things. Naming the head of government is one such duty, calling elections is another, formally enacting laws is a third. The Head of State – monarch or parliamentary-democracy president – will normally never act on his own authority, but rather on the ‘advice’ of the elected leaders of the government, which is tantamount to marching orders to him.
The key word there is ‘normally’. Stable government in a parliamentry democracy is a matter of custom and lubrication, not of conflict. In the US, Congress debates whether to pass a law the President wants, and if they pass one he doesn’t want, he can veto it. In the parliamentary system, by the time it gets to the monarch, it’s already been negotiated out, behind the scenes, and his signature enacting it is a formality. And the monarch retains a bunch of powers, called the royal prerogative – which are used normally only on the advice of the Cabinet. In the UK, for example, the monarch declares war – but he’ll do it only on the advice of the Cabinet. The monarch retains the power to veto a proposed law – but any law reaching him will have been passed by the same majority in the legislature that the P.M. controls, so the P.M. would never advise a veto.
It’s times of constitutional crisis when the reserve powers of the monarch come into play. Twice in the 20th century, when Eden and Macmillan resigned, there was no clear leader with a majority in Parliament to become the new P.M. And Queen Elizabeth chose the new P.M. both times – getting advice (not the formal ‘advice’) from astute leaders both times, to be sure, but it was her decision. Also, what happens when the P.M. loses a vote of confidence is instructive: the P.M. has the free choice of whether to resign (and let the opposition try to form a government) or ask the monarch for a new election. But, first, if he asks for an election, it’s the monarch’s choice whether to grant it. If it’s been less than a year since the last election, he would likely feel free to refuse it – forcing resignation and giving the opposition a chance to try to govern. And, second, though this has never happened, it could: if the P.M. chose to neither resign nor ask for an election after losing the confidence of the legislature, the monarch has the power to dismiss him – because that’s a clearcut power grab against the popular will.
Finally, those weekly meetings of monarch and P.M. are a bit more substantive than they’re made out to be. The monarch has the right to be informed of what his government plans to do in his name, yeah. But take the present situation in the UK: the Queen has been in on every substantive government action since 1952, and knew something of what was going on from the start of WWII on. That’s an immense repository of inside knowledge, available to the P.M. from the one person in the country with it whom he knows is not after his job, who would neither benefit nor be harmed from his staying in or losing office. One would have to be either a complete idiot or an egotist on the order of Maggie Thatcher to ignore that resource – and the P.M.'s of the last 50 years have not been complete idiots.
Finally, there are occasions – rare but real – when the monarch speaks with the voice of a united people. Honoring a public hero is one common use of that – but the best example is Norway in 1940. King Haakon VII unilaterally defied and denounced the German invasion, and set his people to resist – and was supported by the entire Norwegian populace, except the 30,000 Norwegian Nazis. This is the shining example of a Head of State giving voice to the unified feelings of his people.
Figurehead? Yeah, most of the time. But always? Nope – the Head of State acting other than formally is like impeachment or constitutional amendment over here in the US – done rarely, but there when it’s needed.
This is no longer the case.
Since the summer of 2006 when the Constitutional Reform Act 2005 came into force:
The head of the judiciary is The Lord Chief Justice (currently Lord Judge (yes, really)),
The presiding officer of the House of Lords is The Lord Speaker (currently Baroness Hayman), and
The title ‘Lord Chancellor’ is now given to the Minister with the cabinet post of Secretary of State for Justice (currently Jack Straw) - Lord Chancellors will be MPs instead of peers from now on.
Of course, thanks to the Filibuster Eternal, it’s actually 60 Senators.
Cite?
While I thoroughly concur with everything else you said, that is a slur on Margaret Thatcher. Can you cite an example where she ignored the advice of the Queen? Please note that acting contrary to advice is not the same as ignoring it.
As other posters have said, monarchs have a lot of power in theory. Like the King of Italy, he was able to work against Mussolini. Which, as even Hitler noted, was a mistake not to remove the king when he held power.
In Fiji, which is about evenly divided between native Fijians and Indians who immigrated from India when Fiji was a British colony, had a crisis (the Indians managed to form a government over the natives), the natives pulled a coup, and Queen Elizabeth sided with the Indians, in a rare show of political muscle. Of course, the native Fijians just removed her as Queen of Fiji and became a republic.
So that is what shows happens when you exercise your theoretical powers. Despite what others think of the royal family in the UK, Queen Elizabeth commands a HUGE amount of respect. She’s pretty much spent her entire reign without missing a beat, so anything she voices will be given much consideration. Because any politician who goes against her, will have to deal with his constituents. Whether or not any other royals will get the same level of respect as Queen Elizabeth remains to be seen.
Strictly interpreting what he said, he didn’t say that Mrs. Thatcher ever ignored the Queen’s advice, just that she had a big enough ego that she might have.
As for “acting contrary to advice is not the same as ignoring it”, I wish my first ex-wife had believed that. It would have saved me a lot of nights sleeping on the couch… 
Cheers,
bcg
It’s true that Polycarp didn’t actually say that Thatcher ignored the Queen’s advice, but I think that the evidence out there is that Thatcher didn’t let her ego get in the way of her feelings 'bout the Queen. Per Wikipedia, she had this to say about ERII and her weekly meetings:
King Juan Carlos of Spain is an interesting outlier, in that he held serious power on his ascension to the throne and pretty much used it to install constitutional monarchy in Spain, staring down a military coup in the process.
Margaret Thatcher always struck me as the PM of recent times who was most cravenly respectful to the Queen. Maybe that’s why she and HM didn’t get on, if that is true (it’s not something I’ve heard from any authoritative source). As for her being an egotist… well, she was Prime Minister. You have to have a fairly high opinion of yourself to run for that job in the first place. I would say that if anything she was less self-obsessed than… ahem… certain other recent incumbents. She doesn’t seemed to have planned her ascent to the top, unlike them.
This makes me wonder: wealth. In the U.S, the government finances the transportation, protection, and living expenses of our Head of State. Does this occur for all Head of States in European countries? Can the King of Spain or Queen of the Netherlands draw upon the nation’s treasury to pay heating expenses or purchase a new livery for the groundskeepers? Do they draw a salary?
In Denmark, the upkeep of the royal house is part of the state budget. (It’s called the Royal “Apanage”.) Palaces, crown jewels etc. are part of a cultural heritage quite apart from them being used in the running of the country, so it’s not as if it’s an undue burden. Her Majesty allocates the funds as she sees fit, has people to hire and fire etc. etc.
In the UK, the expenses needed to maintain the monarchy are paid by the state. See Civil list. The royals also own a substantial amount of property themselves, so they are still very wealthy.
Plus they get to print their own money. It’s got their picture on it and everything.
When she was but a mere MP she was quoted as predicting that there’d never be a woman Prime Minister in her lifetime. Heheheheheh. You’ve got to love that kind of Real Life irony.
Cheers,
bcg
A president can’t pass a law, but they can sign executive orders which are pretty powerful in some areas. For example next week Obama is going to allow federal funding of stem cell research after Bush banned it.
As far as funding the royal family, about five years ago Queen Elizabeth started paying taxes voluntarily, I guess she is not required to by law.
I think that is definitely something that would require the full amending formula.
Also, I don’t think Canada can even do so – doesn’t the Statute of Westminster require all Commonwealth Realms to assent to changes in the succession? (I mean, theoretically we could cease to be a Commonwealth Realm and set up our own royal dynasty with whoever we wanted as monarch, but that seems like an awful lot of trouble.)
Which is ironic since technically she’s paying taxes to herself. Only the Queen and Prince Phillip have a Civil List income in their own right, the other royals get an allowence from the Queen’s portion. All except Prince Charles who derives his income from the Duchy of Cornwall.
As I understand it, taxes go to The Crown, which is distinct from The Queen herself. So Her Majesty isn’t paying taxes to herself, she’s paying taxes to The Crown, which can best be described as a sort of legal concept- a “Corporation” or “Company”, if you will, representing the authority of the Government to exist and carry out its functions.