Let say I have 3 shapes in illustrator: A, B and C. I would like A to overlap B, B to overlap C and C to overlap A. How can I do that?
You have to "cheat". If it's a simple enough object, I would just duplicate shape C, bring it to the top, and use a Clipping Mask to cover up everything but the intersection.
To do so, create a shape with no fill or stroke around the intersection (can be any shape but I'm using a rectangle for demonstration):
Select both your empty shape and the duplicated object that should overlap and do Object → Clipping Mask → Make
Another way you could achieve this overlapping effect would be to use the shape builder tool. You would create your shapes in the position you would like. I will use the example like @JohnB's.
Keep in mind that using this method will break apart the original shapes and you should consider this in case you want to revisit positioning at a later time, etc. In that case @JohnB's approach would best fit because it retains the integrity of the shape within a clipping mask.