Watched all the Giant Bomb Chrono Brono Trigger Endurance Run. (Well, I fast-forwarded some parts, but I still watched every episode!) Never had played the game before, only heard about its greatness.
I'd maybe pick it up for DS or something eventually just to play through my own way. Multiple Endings and New Game+ would keep it fresh enough.
That's the only play through of a game I can remember completely watching from beginning to end though.
Other examples that come to mind are having watched a good bit of a Let's Play of Amnesia, and a decent bit of the original Deus Ex (game is a bit dated now - I do have HR in the backlog via Amazon/Steam Sale, but I may wait until I get a more powerful computer to play it - a bit unstable on my laptop.)
I know I've watched at least portions of other Let's Plays, and certainly smaller bits and pieces of various games here and there.
If a game is already out, I find that Let's Plays are a really good way to get a feel on whether I'd enjoy a game or not. Especially if a game doesn't have a demo (which aren't always the best indicators of final game experience/quality anyway.) Besides using Let's Plays to gauge a prospective purchase, they're also fun to watch for old games I'd be unlikely to get ahold of, or even play in the first place. Or just to revive some memories for games I have put away somewhere without dragging whatever system out of storage. And since you can fast forward, you don't necessarily have to play through dull sections just to get to interesting parts.