Two books to add to the already comprehensive list. However, these books focus less on building a rigorous treatment of topology and more on building intuition.
The first, for geometric topology, is A Topological Aperitif. I read this book a while back and really enjoyed the geometric flavor, and it introduced some useful concepts for topology.
The second, for algebraic topology, is An Intuitive Approach. This books covers almost all of the major topics of algebraic topology, using very intuitive explanations in about 140 pages. Most proofs are left out, but intuition for this material is very nice. He covers all the basics except covering spaces and higher homotopy groups.
Just my two cents.
