I'd have to re-watch the EEs of the first two films to pick one from each (there's a good excuse)
However, for Return of the King I'd have to give a mention to the scene where Aragorn, Gimli and Legolas stumble out of the Paths of the Dead and Aragorn sees the black ships. I had to explain
how they had got onto the ships so many times to people who had not read the books
My favourite additions though, were the scenes in the Houses of Healing (all of them) I really regretted their absence in the theatrical version.