Big things are happening on TV Tropes! New admins, new designs, fewer ads, mobile versions, beta testing opportunities, thematic discovery engine, fun trope tools and toys, and much more - Learn how to help here and discuss here.
The Bottle Dance, for the sheer amount of balance required to pull that off. Most productions use trick hats.
You'll notice, that the first part of that sequence is all one shot, which is mad impressive.
That dance is still done as "Shtick" (dances done in order to please the Bride and Groom) to this day at some Jewish weddings. (specifically, at this troper's parents wedding... where there was a guy balancing a bottle, a cardboard, and 4 bottles on top of the cardboard)
Most productions use trick hats. This Troper attended her sister's middle school production, where six girls did the Bottle Dance with no tape, no string, just flat-topped hats and bottles. In three performances, only a single hat was dropped, even when they went from standing to kneeling and back!
Some of them have even found the dance is easier without a trick hat. You just wear a hat big enough, push in the top, then rest the bottle in the dent. The tape to hold the bottle in place often isn't strong enough anyway.
The whole soundtrack, but especially "Sunrise, Sunset".
"L'Chaim (To Life)" also deserves a mention. It can be viewed here. Not only will it make you want to get wasted, but the Cossacks breaking it down is nothing short of fantastic.
Dark Reprise - The chorus comes on and sings "Tradition" after Tevye disowns Chava.
Just the song itself is awesome.
The Broadway revival (at least) had the chorus reprise the dance of "Tradition," too - only now it's a dance of entrapment and anguish from which no-one can escape.
Tevye intimidating the constable after receiving the news that they have to leave. "This. Is Still. My land. Get. Off. My land."
"Even a poor tailor is entitled to some happiness!"