These are all expositions of topics that I am interested in. I'll attach a note if I have actually read the thing (sometimes even if I haven't.) Suggestions and reviews are welcome .
Rational Tangles are one of Conway's many fascinating discoveries (a basic introduction can be found here.) This paper by Kauffman proves the main theorem in a self-contained manner.
The Banach-Tarski Paradox A truly bizarre consequence of the axiom of choice is that you can take a solid ball, cut it up into a finite number of pieces and then reassemble those to get two balls of the same volume. This is a very nice and detailed exposition written for the Harvard minor thesis. It is readily accessible to advanced undergraduates.
A special case of Sharkovskii's theorem Let f be a continuous map from the unit interval to itself. A point is said to have period p if f^p(x)=x and p is the least integer for which this happens. This paper proves a special case of Sharkovskii's theorem - if f has any periodic point that is not fixed, then there exists a point of period 2. The paper is dense at first reading but the method is quite interesting.
The 'opposite' special case is that if f has a point of period 3 then it has also a point of period n for arbitrary n. That is the famous "Period Three Implies Chaos" theorem.
The Lester Ford award is given to upto five outstanding expository papers every year. You can find the awardees here. Big names on that list include Kac, Halmos and Milnor.
E-mail me with more suggestions at abhishek at ocf dot berkeley dot edu