Ragtime - "Prologue" - Others have explained why it's so wonderful, but I absolutely love the part near the end when it picks up: "It was the music of something beginning: an era exploding, a century spinning..."
Parade - "The Old Red Hills of Home" - So many composers would have started off the show a different way, but JRB creatively chooses to go all the way back to the Civil War. Awesome. Also, I think it sets up the whole conflict: North vs. South, innocence lost, the Southern way of life seen as ideal, which the North of course changes.
Sweeney Todd - "The Ballad of Sweeney Todd" - Probably the coolest beginning to a show, ever.
Les Mis - "Work Song" - I remember when I saw Les Mis, sitting there in the dark, waiting for the show to begin, and all of a sudden I get jarred by the opening notes. The effect was incredible.
Evita - "Requiem for Evita" - It's a bit weird when you're seeing the show, but it's become one of my favorite parts of the score. I love the contrast between the very heavy, sad "Requiem" and then later on the lighter "Oh, What a Circus."
Side Show - "Come Look at the Freaks" - I haven't seen Side Show, but I wasn't really looking forward to listening to it until I heard this opening, and then I knew I was in for something good.
Songs for a New World - "The New World" - Again, haven't seen this show, but I can only imagine what it's like performed live.
The Light in the Piazza - "Statues and Stories" - Absolutely beautiful
Miss Saigon - "The Heat is on in Saigon" - Definitely gets the audience's attention, though I like the overture more.
I don't think these are the best, but "Jailhouse Rock" from All Shook Up and RENT's beginning (especially the title song) are absolutely electrifying when you see them live.
Jimmy, what are you doing here in the middle of the night? It's almost 9 PM!