Suppose Will gets Kate pregnant, and then they take a short trip to, say, Aspen. For purposes of the hypothetical, lets assume that it is a strictly private visit, and they are not there in any sort of official capacity. Lo and behold, the baby comes a little early, and they are suddenly the proud parents of a little prince who just happens to have been born in the United States. Presumably that would mean he is a US citizen. Assuming he never renounces his citizenship, would he still be a US citizen when he ascends to the throne? Or, would this be considered a constructive renunciation of citizenship?
I don’t see why it should be a problem theoretically - I know the US can’t recognise titles and all that jazz but it isn’t going to be them giving him them. And don’t forget you gave Winston Churchill an honorary US citizenship - I cannot remember his position but I’m sure he was in the first five.
The prince however would renounce citizenship if he had any sense given the way you, very rarely for a country, tax folk on their worldwide earnings*
*Acutally, so does my little country - but we cap income tax at £115k cause we don’t like to steal from people as much
it would be more amusing to see if the child would decide to run for president!
If they can crown Germans, they can crown Americans.
First five of what? Winston Churchill was, at various times, Prime Minister of the U.K., so he was the leader of the government, but he wasn’t royalty or nobility (though the grandson of a duke), and he wasn’t in the line of succession to the monarchy.
To answer the OP: there’s no restriction on citizens or subjects of foreign powers being the monarch of the United Kingdom. The only negative qualification is that you must not be a Roman Catholic. So if the Duchess of Cambridge gave birth to a child while in the U.S., that child would be a dual U.S./U.K. citizen, and would be third in line to the U.K. throne (after its grandfather and father).
I believe the OP is asking the question from the point of view of U.S. law, not British.
Right, from the point of view of US law, would a US citizen becoming a monarch of a foreign country lose their US citizenship? I think the question is probably academic, because a foreign monarch is unlikely to try to use any of rights of a US citizen. They would not vote or run for public office in the US, and when they visited the US they would not be forced to choose between the immigration lines for US citizens and for others, because foreign monarchs don’t go in those lines.
However, what happened when Grace Kelly became princess consort of Monaco? Did her joining a foreign royal family cause her to lose US citizenship? I don’t think so: the Wikipedia article says: “She also retained her American roots, maintaining dual US and Monégasque citizenships.” If Her Serene Highness could do that, I don’t see why an American-born U.K. monarch could not as well.
Sure.
edit: sorry not British
First five to be given it
No, as far as U.S. law goes, it’s an exact match: His Majesty the King of Thailand would be a US citizen by birth, and as far as US law is concerned he is the monarch of a foreign nation. I doubt if any US government official has declared that His Majesty has lost his US citizenship just by becoming king: there would be no point, and it might be offensive to the Thais. On the other hand, he may have renounced his citizenship.
I had heard that only he and Lafayette had the honor. Who are/were the others?
According to Wikipedia:
Raoul Wallenberg
William Penn
Hannah Callowhill Penn
Mother Theresa
Casimir Pulaski
Winston Churchill was the first, in 1963. With Lafayette, a total of only seven.
What a silly list! I hadn’t seen the entire thing before. I’m only familiar with a few of them, but I know Mother Theresa was a cunt, William Penn was dead a zillion years before the US, Hannah Penn was his wife and therefore even more pointless by this scheme. Raoul Wallenberg was a wonderful man but frankly his Swedish citizenship makes giving him a US one superfluous; it is not as if he aided US interests in the second world war - he aided the interests of decency and humanity but these are not the same things. And sorry to say, I don’t know who the other one is and while I will google (I think) her after this it would be wrong for me to comment on (I think) her now.
I continue to read. But don’t I look silly 
Not necessarily. Given who his parents were, very likely they were in the US on diplomatic passports. Children of diplomats are not automatically US citizens by birth. While it’s possible they were here under regular visas, that’s not the way I’d bet. I would reserve judgment until I see a cite that he was a US citizen.
Interesting. If say Putin’s wife had a child in the states, could they opt to have dual citizenship for the child? (quick hijack)
What if someone in the US on a diplomatic passport has a child with a US citizen, rather than both parents being on diplomatic passports?
Source of a day off for school kids and public workers in Illinois. 
More seriously and completely: Polish nobleman, served as a general in the Continental Army for the Americans in the Revolutionary War. Due to the large population of Polish-Americans in Chicago, his birthday is a civic holiday in the state.
His parents were students at colleges (Harvard, MIT and Simmons) in the Boston area. As members of the Thai royal family they may have been entitled to diplomatic passports, but they were neither of them diplomatic or consular officials at the time.
Lafayette was already a citizen of the United States. He had been given citizenship to the state of Maryland in 1784, so when the Constitution was adopted in 1787, he automatically became an American citizen.