I believe you're not allowed to renounce your citizenship if you don't have another one; in other words, you can't be stateless.

Edit: Interestingly enough, the US allows it. The Wikipedia page on statelessness says "The United States, which is not a signatory to the 1954 Convention on the Status of Stateless Persons or the 1961 Convention on the Reduction of Statelessness, is one of a small number of countries that allow their citizens to renounce their citizenship even if they do not hold any other." See https://en.wikipedia.org/wiki/Statelessness

Right, but I don't know what happens when you are a "stateless" person. Are you ever allowed in ANY country? Are you deported from the US after becoming stateless? I have never seen a good explanation of what happens AFTER becoming stateless.

I would guess that this varies a great deal depending on what country one happens to be in at the moment of becoming stateless, but that's really just a hunch.

Update: the Wikipedia article seems to back that up.

https://en.wikipedia.org/wiki/Statelessness

These are people with double citizenship.