I've always found self-referential paradoxes fascinating. The most obvious one is the Liar's paradox, like the title of this post. Another I like is "this sentance has threee errors" (the third error being that it has only two. But wait, that means...).
Then there are the Godel-style ones which demonstrate the limits of logic. Consider the theorem G: There is no proof of G*. Is G true? Then it's not a proveable theorem (and with a bit of extra work we end up having to conclude that mathematics is incomplete). Is G false? But then we have a false proof (so mathematics is inconsistent)!
* = I've simplified that a bit, what we really want to consider is the theorem G: It is not the case that there exists x and y such that (x is a proof of y, and y is the Godel number of G), expressed in the language of arithmetic.
Or in metaphysics, realists about 'properties' have to face various problems regarding exemplification. [Quick background: an object is said to 'exemplify' a property if that property is possessed by the object - e.g. red objects exemplify the property of redness. A property is said to be 'self-exemplifying' (SE) if it exemplifies itself. For example, the property of 'Being a property' is SE, but 'being triangular' is not, since properties themselves have no shape.]
For consider the property of Being Non-Self-Exemplifying (BNSE). Does this property exemplify itself? If so, then it has this property: namely, being not SE - so it does not. If not, then it's not SE, so it has the property of BNSE, so it is! Either way, we have a contradiction.
Feel free to mention your own favourite paradoxes of this sort in the comments.
What I've just been thinking about though, is the Rochester philosophy blog, whose name is "This is Not the Name of this Blog". At first I thought it was another nice paradox, but now I don't think it works at all. Instead, it's just plain false. For it is the name of the blog, and in asserting that it's not, it asserts something which is false. But there's no paradox there - its falseness does not imply its trueness, or anything interesting like that. So that's a pity.