My favorite scene from all three, comes in TTT...the scene between Boromir, Faramir, and Denethor in Osgiliath. One mainly because Boromir is my favorite character, and it shows the friendship between the two brothers, despite Denethor's favoritism (and some wonderful lines).
"Remember this day brother, it is a good day."
"...that leaves more time for drinking," (Ok I forget half the quote).
"He loves you father!"
This scene also goes to explain the actions of Faramir, taking the ring to Osgiliath.
From ROTK, my favorite added scene would probably be Eowyn fighting fiercely next to Theoden. I love the expression on Hill's face like..."who is this soldier?"