The I'm lying paradox is a simple one.
Let's say that an American says: "All Americans are liars.".
This means that the person was lying when (s)he said that, because (s)he is an American.
This means that all Americans tell the truth, which makes the original statement true, and the loop repeats.
There are a lot of paradoxes is this category, and I will post more of them to this website as time goes on.