For me, there would be two: Fellowship of the Ring and War of the Ring.
FotR follows the book rather closely, despite some of the sillier moments and it's dated, Elder Scrolls III: Morrowind style graphics (except for faces). Also we get the noted voice actress Jennifer Hale as both Galadriel and Lobelia Sackville-Baggins. I feel as though this version of the story should be concluded with next-gen graphics, especially since it's been long enough that some of the hype from the movies has died down. It would be a great time to bring this series back, with The Treachery of Isengard and The War of the Ring, where we get to see the rest of the book's adventures played out.
WotR is very similar to another nostalgic RPG/RTS, Warcraft III: Reign of Chaos, but it's still very fun to play. It only has two sides and a decent campaign, though the lack of the Siege of Minas Tirith and the Battle of Pelennor Fields is a disappointment (but playing the siege of Osgiliath with Boromir and Faramir and the Battle of the Hornburg are certainly high-points). I've also seen that some of the Edain modders must be in some way fans of this game, as I've recognized bits of the dialogue of the Elf Lightbringer and Goblin slave-master in 3.8 and 4.0 (though in German). An expansion that would be bigger (especially with more characters for the Battle of the Pelennor Fields maps - think Eowyn having similar abilities to Boromir, Theoden being fleshed out, Eomer and Imrahil both being DPS/Tank style cavalry heroes, and maybe even more units like the Knights of Dol Amroth and Variags of Khand - i love me the men of Dol Amroth) would certainly be welcome, though it's never going to happen, as the game is so old that Sierra has likely forgotten about it.
How about you? What are your favorite non-Jackson Lord of the Rings games?