I think the simplest, and probably the easiest to understand in its impossibility, is the following:
The sentence below this one is false.
The sentence above this one is true.
Simple, but quite a mindfuck if you really follow it around for long enough in that loop. The infinity of impossibility is really emphasized here in a very simple way.
Overall, I think that paradoxes underline our ability to comprehend that there are things we're not able to comprehend, but that we are also tenacious enough to try and solve the unsolvable anyway. That, in itself, is the real paradox.
Thinking about the prime mover is one of my favorites. "What created the thing that created everything?" The question itself, logically, is infinite and a paradox. If we are to logically assume that in order for something to exist, it must be made to exist by a previously existing force, then our entire understanding breaks down at a point if you're rewinding in causation. But all this assumes we're operating under a unified logic, which is mostly the case.
The intriguing question is whether the paradox is unsolvable, or if we're putting square pegs in a round hole because it's the only hole we see.
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