The United States has no prohibition against acquiring other citizenships. According to the
U.S. State Department website:
"A U.S. citizen may naturalize in a foreign state without any risk to his or her U.S. citizenship"
I have a work colleague who has Lebanese, Armenian, and U.S. citizenship.
The limiting factor will be the citizenship laws of other countries. Some do not allow it at all (China and Japan I think are a couple). The rules for citizenship
by descent varies by country. Ireland is comparatively liberal in allowing the grandchildren of Irish born emigrants to request it. Other countries only allow the children of their emigrants to do so. I think Britain is one of those.
Some allow multiple generations removed from emigrant's home country. I believe Greece is one. Italy apparently allows it for several generations, but it is apparently a very slow process (2-5 years). There's also some weird rules about whether or not an ancestor naturalized in the US or renounced Italian citizenship. It doesn't apply to me, but I have a friend who did some research.
In addition to two Irish grandparents, I had two Polish grandparents. Poland does allow you to claim citizenship through grandparents, but only if they left Poland after it became an independent country in 1918. Mine left around 1905. Regardless, I think it is a more complex process that would require getting an attorney in Poland to help, and to have all your documents translated into Polish. Anyway, both Ireland and Poland are EU members. So a Polish citizenship and passport wouldn't seem to offer any incremental benefits. And I don't speak Polish.
I believe that anyone who is Jewish has the "Right of Return" for Israel, which would allow them to apply for citizenship.
Some other things you might have to watch out for are things like mandatory military service or other obligations in another country. Taxation could be an issue. Generally, the US requires its citizens to file/pay taxes in the US, no matter where you live. If you spent most of your time in Ireland, you would still have to file US tax returns. The US does have tax treaties with Ireland and other countries. So, I don't think you'd pay more than whichever country charges the most. If you want to eliminate your US tax obligations, you'd have to renounce your citizenship, which costs around $2,400 (maybe more). If you do that and want your citizenship back, you need to apply for residency, just like every other foreigner.
This is an agglomeration of information I picked up perusing the web. But if anyone is truly interested in citizenship in another country, please don't take what I write as definitive.