More or less every final boss mini-game is tense enough to work as one whenever the player wins.
The victory skits at the end of each board in Mario Party 2 has the top player heroically defeating Bowser in a way that matches the theme of the board, which is especially fun after all how much he messes with you during the board. The one for Bowser Land takes the cake, though.
Winning a particularly tough board or mini-game, or achieving some other great advantage to put to good use, can make a player feel awesome.
Donkey Kong punching Bowser in Mario Party 5 on the occasion. I repeat! DK! PUNCHING! BOWSER!
Daisy slapping Bowser into the sky in Mario Party 3.
Waluigi gets his turn when he beats Bowser up later on.
The Frightmare Bowser fight in Mario Party 5, especially the last phase—and themusic.
This ENTIRE video for Luigi! Takes place in Mario Party 9.
In Mario Party Island Tour, in Story Mode Bowser wins. When the character reaches the top of Bowser's Tower, Bowser will reveal that the destroyed Bowser was a decoy and then proceed to knock the character off the tower, saying that they can come back any time.
Mario Party 10's star is Bowser, who is playable for the first time (not counting a bonus game in 4). Whoever plays as him can ruin the other players' days.