Boy, these movies are filled with great one-on-one fights. From Saruman/Gandalf, to Sam/Shelob, to Frodo/Gollum, and all the fights in-between, there are many very satisfying confrontations. Before The Battle of the Five Armies came out, my answer would have been Aragorn/Lurtz from the end of Fellowship. But Battle did come out out, and now, of course, my favorite one-on-one fight is Thorin/Azog.
This is also a nice long showdown, for once. And when they finally get out on the ice, alone, it just gets even more satisfying, right down to the conclusion. This one-on-one fight makes me very happy to watch.