dtgreene: Also, feel free to mention games that you keep restarting *before* you finish them.
Some RPGs mainly, because I felt I made some bad decisions either in character creation, or later in the game. Like:
Baldur's Gate: I restarted it several time from a scratch, sometimes because I felt maybe I could have a better character, but sometimes simply because I hadn't played the game for quite some time that I had forgotten what I was doing in the game and where I was going when I left the game. So I restarted the game "more organized", keeping a log myself which subquests I've already finished etc.
Fallout Tactics: Quite late in the game I learned I had made some choices that were anything but optimal. I restarted it from scratch when I had only few missions left, and yes it did make a difference, my new party grew considerably stronger as I made better decisions in developing them.
Dungeon Siege: Legends of Aranna: I restarted both the base game and the expansion. The base game I restarted because I felt I incorrectly tried to make my characters "jacks of all trades", when the game actually favors strongly specialized characters that concentrate on doing one thing good.
The expansion pack I restarted because at first I didn't understand fully what the "item sets" were all about, and I had foolishly sold some such special items, unable to recover them later (it seems that if you sell some item to a storekeeper, that item will vanish for good after awhile; so it isn't like Fallout 1-2 where the storekeepers seem to keep everything you've sold them for the rest of the game).
It wasn't a game-breaking thing that I was missing some such special items, but I just wanted to see how well such item sets work together when you find all the pieces. Some of the item sets were quite useful and powerful.