A user just asked this question but removed it after it was heavily downvoted.
Then he proceeded to post the same question again with no apparent change whatsoever.
Is this allowed? It seems like it shouldn't be.
I don't think it is allowed. Even if it is allowed, people will probably come and downvote the question even more.
Technically, it is probably allowed (I don't think that there is a specific rule that prohibits you from doing so). However, it is of course not encouraged and frowned upon by the community.
But I think we should take a bit of a liberalistic standpoint here and let the 'market' do its work. If the question hasn't been altered at all, I don't see why it would not again receive lots of downvotes, and it will in fact probably even get more downvotes because of people like you that will notice the repost and that will (rightfully) downvote. So I don't think there is reason to intervene here.
For what it's worth, the person under discussion at http://tea.mathoverflow.net/discussion/1187/extending-from-a-plane-in-r3-again-and-again-and-again did that for a few years on both MO and MSE, under multiple user names. His final few posts involved advice on job applications; my conclusion is that he stopped only because he got his Ph.D., although that still leaves some questions. Very hard to deal with a really determined cheat, especially as there are generally no real-world consequences for being caught. One girl involved in high-school competitions did get uninvited to a summer math camp after using MSE to cheat, under many names. Not sure of the degree to which self-deletion played a part in that particular story.