My Fair Lady's overture successfully captures the feel of the show, as does the "entr'acte". They're both tuneful and fun. I love them.
I'm partial to Camelot's majestic overture, and the Carousel waltz is brilliant.
For an exciting beginning, I don't think anything can compare to the original (NOT the current tour's) end of The Phantom of the Opera's prologue, followed by a loud bang, and the chandelier wiggling about and starting to rise as the music plays. I'm not sure what the music is called when it comes after a prologue -- I've heard it's not technically an overture. BTW, POTO's entr'acte is pretty darned impressive, too.
Audrey, the Phantom Phanatic, who nonetheless would rather be Jean Valjean, who knew how to make lemonade out of lemons.