It’s easy to have three citizenships at birth - one parent from country A, one from country B, born in country C - but can you have more? 4 might be possible if where you were born was disputed between two countries, for instance. How about 5? More?
AFAIK, any Jew can claim citizenship in Israel. So that could be the fourth.
You might also be able to claim more if your parents were themselves multiple-citizenship.
Sattua, I think that, while any Jew can claim Israeli citizenship, it’s not automatic. You need to go through the paperwork to claim it, which would take some amount of time. And I doubt that the process could be initiated for a child before birth, so the kid woulnd’t have Israeli citizenship at birth, as the OP asks.
Suppose you had one parent from country A, one from country B, and were born in the embassy of country C to country D. And were Jewish, so could claim Israeli citizenship as well (and Israel was none of countries A-D)? Wouldn’t that give you five citizenships?
Most countries don’t grant citizenship to everyone who happens to be born on their soil. And embassies are the soil of the host country, not the sendint country.
If that’s so, there’s no theoretical limit to the number of citizenships you could have (until you ran out of countries to be a citizen of). Just repeat the process for a few generations. For example - If both your grandfathers had three citizenships each, and both your grandmothers had three citizenships each, this gives each of your parents six apiece and you have twelve at birth (assuming no duplicate citizenships among your grandparents).
I feel there ought to be something wrong with this argument, but I’m not sure what it is.
Didn’t Ireland allow the (grand)children of citizens to obtain citizenship? They stopped this loophole in the 90s.
You might run into problems with residency requirements for citizenship by descent. For example, a child born to a U.S. citizen and a non-U.S. citizen is a U.S. citizen at birth only if his or her American parent spent at least five years in the U.S. before the child was born, and at least two of those years were after the parent turned 14. I could envision an extreme case where a young parent had a large number of citizenships, but because of moving around from country to country hadn’t spent long enough in any of them to “pass on” the citizenship.
Laws concerning citizenship by descent are different for each country, though, and you can’t really make blanket statements about what’s possible. It might be an interesting exercise to try to construct a scenario which maximizes the number of citizens of a child born in the present day; but to do so would require more research than I’d care to do. 
Any Jew can claim Israeli citizenship, but they have to move to Israel first. You can’t become an Israeli citizen from Teaneck, NJ.
Didn’t Israel go ahead and grant citizenship to one gentleman at the time he was incarcerated in a US federal prison?
You mean Pollard? I’m pretty sure he wasn’t awarded citizenship, and even if he was, it was a very special case. Political.
People are assuming that in general, countries bestow citizenship by descent automatically. It’s not the case, for example:
I think Italy still grants citizenship this way: if you have an Italian ancestor, I think, you can claim Italian citizenship. I don’t have any cites, unfortunately, only the recollection of how many foreign football players in the 80s would mysteriously acquire nebulous Italian ancestors when hired to play for an Italian team, so they could be allowed to play in the Italian championship, which only allowed two foreign players per team.
So if this is true, someone could add an Italian citizenship to the list of the ones they hold from their parents’ if they have an Italian ancestor. Let’s imagine a descentant of a Roman Jew who migrated to, say, the US…
Many countries do not grant automatic citizenships to babies born in their country if the parents are not citizens. Japan does not, so our children will be dual citizens of Taiwan and America, but not citizens of Japan. As mentioned above, there may be some case where you can get multiple citizenships, but this isn’t as easy as the OP suggest.
I know the OP said at birth, but if the primary citizenship is the US, then expect to be hounded until giving up the others. The US “discourages” dual citizenship. US embassies at least in China can be real SOB’s about this. So, you might be born with several, but after a few years of active discouragement from the US you might end up with just 1.
Yes, it was Jonathan Pollard, and yet, it was a very unusual case.
And China does not allow dual (or multiple) citizenship, so strike China off the list when researching. My daughters, born in China, had to give up their Chinese citizenship when they became US citizens.
I wonder if this is because the dual citizenship was with China? My boys are both Australian/American dual citizens and we haven’t had any comment made from either country, even through all the visa stuff we had to go through at the U.S. embassy when we moved back to the U.S.
Also, I’m wondering how many countries consider a person a citizen at the moment of birth. My son who was born in the U.S. wasn’t considered an Australian citizen (even with an Australian father) until we applied for citizenship. My son born in Australia had to have his birth registered with the consulate, but I believe he was considered a U.S. citizen from the moment he started breathing air.