I've done this with my Spanish passport entering Brazil. No visa required, and a Spanish passport is cheaper, faster, and easier to obtain than a Brazilian visa (or even a US passport as a natural-born US citizen).

The only annoyance was a CM check-in agent at SJU insisting that Spanish citizens require a visa like US citizens, but that was easily resolved.

Not sure what this means, the US does not have any exit controls, you don't show your passport to any authorities when you leave.

US law states that US citizens must use their US passport to enter the US, and must have it in their possession when leaving. This doesn't necessarily mean that the US passport must be used for the departing flight. The fact that there isn't an exit control under normal circumstances doesn't mean that there can't be requirements placed on exit. For example, there usually isn't a Customs inspection upon exit, but there certainly can be. US Customs can and does appear occasionally on the jetway for departing int'l flights and conducts inspections - they can go through all your stuff and search you without cause just like they can upon arrival.

With all due respect, you sound quite naive as to the requirements for passport issuance. Governments don't just hand out passports to non-nationals just for the hell of it, orr because they ask really nicely, or because they're American.

Almost every country's process of acquiring citizenship (unless it has been conferred by an ancestor, and even it is is rather restrictive in most cases) is long and expensive.

If you're worried about being at a disadvantage in a hostage situation, just remember that there are tens of millions of US nationals overseas at this exact minute in virtually every country in the world, and your chances of being singled out because you are a US citizen are literally infinitesimal.

Relax and enjoy your trip.

Renunciation of US Passport/Citizenship is an option that is worth serious consideration...

Unless you have a grandparent from a country which allows conferral of citizenship to grandchildren (Ireland, Italy and Lithuania are the three I can think of, but there are surely others), the answer is no.

I'm a bit curious as to why you think any EU country would just give you citizenship if you just turned up on their doorstep. It hardly bears mention that the US certainly does not do this. Why would any other country?

Looks like a chink in the Brazil visa "armor".
More exceptions will come with time
from those who have the knowledge,
one or more of which may apply to my itinerary:

"...you can get into Brazil without a Visa is if you are on land, coming over from Guyana. If you get to the border town of Lethem, you can walk across the bridge and the border guards will let you go over, get a lunch and spend a few hours, but you must walk back over to Guyana by the end of the day."

Odd, how resentment is stirred up in those who obediently paid
without doing research on possible money-saving strategies.

That was the point I tried to make, joking about giving out passports like candies earlier. But OP doesn't seem to get this particular point.

If I am not mistaken, it will cost $450.

Since $450>$420, I doubt OP would consider it saving money. ;)

FT PRO BONO Warning:
 

Before looking at a short term monetary saving in getting other citizenship/passport one should investigate and consider all the long term cost and obligations of acquiring other citizenship/passport one is entitled to.

I think the "hostage situation" came up as a rumored reason for 3rd party countries to issue passports, not as a desired reason by the OP.

Anyway, the issue is that certain countries charge high visa fees to US citizens, and the OP wishes to avoid this. If this were easy, everyone would do it.

You have to either find a country that will admit you as a citizen, or issues passports to non-citizens. The latter is rare, and is usually restricted to refugees.

Unless you do have the right relatives, getting citizenship in another country is going to cost a lot more than the visa fees. Renouncing your US citizenship won't help either; you need to get citizenship in another country that has lower visa fees. Having no passport at all won't help.

Well, it is easy. Just don't go to countries if you don't like how much they charge for visa fees. Problem solved.

And lobby your Congressperson to push for the removal of ridiculous visa fees for nationals of most countries to visit the US, which is a source of most of the reciprocity action behind these fees.

But in many cases that is an actual falsehood. In the cases of Brazil and Turkey, while their nationals required visas, they were free of charge, in both cases it was they who started the visa fee reciprocity wars. Argentina is a bit more complex as there was a period until a decade ago when Argentina was a visa waiver country, after their economy collapsed, their citizens violated their stays in the US almost en masse (from less than 1% overstaying to 40% in less than 16 months IIRC) so the US required them to have visas, they reciprocated and you have the situation that we are in today. India, China and Russia are interesting ones as the Chinese extort visas fees from almost every single nationality, often higher than visas the other way around. India is all about straight reciprocity, but at least they have adjusted it so that the possible validities are the same (they issue up to 10 years for US citizens), while the Russians are the worst as they rarely give more than a dual entry for 3 months, charge a hell of a lot for it, and for longer visas require AIDS tests and all sorts of baloney.

That's putting it lightly. Even if there was no fee to renounce US citizenship, doing so in the OP's situation would be very problematic for several reasons. A renunciation in the absence of another citizenship would make one stateless, which creates several problems:

1. Renunciation must take place at a US embassy or consulate outside the US. As soon as the renunciation was effective, you'd be an illegal alien in the country where the renunciation took place as your US passport you used to enter would no longer be valid.

2. You'd be unable to travel to Brazil, since you'd be stateless, unless you applied for some sort of special refugee or stateless person travel document, which is unlikely to be granted to a previous American that just wants to avoid paying a $150 visa fee.

3. You'd be unable to return to the US for the same reason.

I'm really just astounded that the OP actually seems to think he could just show up in any "EU" country and say "Hi - I'd like to be a citizen, please, because I really can't be bothered to pay entry fees to some countries," and they would just say "Sure! Here you go! Don't forget your passport!", as if it were that easy.