At n = 3, every blue-eyed person knows everything every other blue-eyed person k...

At n = 3, every blue-eyed person knows everything every other blue-eyed person knows and that every blue-eyed person knows it, so you can already construct an infinitely-long chain of "I know that he knows that I know that he knows..." at that point.

(Of course, I am assuming that all the people think/reason exactly the same and that they all know that they think/reason exactly the same, which are assumptions of the original problem.)