The Liar's Paradox
The classic self-referential paradox is, "This statement is false", where 'this' is taken to refer to the statement itself. If the quoted statement is true then it must be false. If it is true then it follows that it is false. There seems to be no way out of this bind. Another variant is, "I am lying now".
If you think this sort of self-reference is not allowed then consider the pair of statements:
A) The next statement is true.
B) The preceding statement is false.
Taken together, A and B form an irresolvable logical paradox. For if A is true then B is true, making A false; and if A is false then B is false, making A true.