Mario's Rainbow Castle in the first Mario Party one has Bowser's event. He gives the player a black star. The music dies out and the character looks at the screen as if to say What. Then Bowser takes 40 coins and does a dance.
Not to mention his event in Luigi's Engine Room. Bowser presents you with his latest invention, a coin-making machine. To demonstrate, Bowser then makes a coin for you, which your character looks at and then looks at the screen as if to say "this can't really be happening" before picking it up. Bowser then charges you 20 coins for the demonstration and does his little dance. Jerk.
In the Mario Party 2 intro, Wario disagrees with Mario Land being named after Mario and declares it be named Wario Land since he's the Super Star. Him, Mario, Luigi, Yoshi and DK all get into an argument, which is halted when Peach interjects and says that she has a name everyone can agree on. She pulls a cord...and a sign for Peach Land shows up. Everyone collapses in response.
Another is the red Koopa Troopa who, upon discovering Bowser invading Mario Land, rushes to the six and yells at them as they're arguing of the threat. They briefly look at him, then go back to arguing as if it doesn't matter.
The music of that bit only makes it funnier. Especially when it abruptly pauses when the cast glances at the Koopa as mentioned and abruptly resumes as they go back to arguing.
In Mario Party 3's Story Mode, Mario is shown happily relaxing outside Peach's Castle in a chair. What could possibly ruin this plumber's day? The Millennium Star crashing right beside him, causing Mario to go flying off.
Daisy slapping Bowser into the sky in Mario Party 3. The reactions of Tumble and the Millennium Star especially make this moment priceless.
Mario Party 3's Bowser in general. He's portrayed as totally incompetent, showing up too late, and getting owned by not only Daisy, but Waluigi, who easily defeats him in a fistfight.
One of the best parts in the Advance version is during the "Bowserstein!" quest. You're given a choice of facing either Bowser's Scary Monster or Very Scary Monster, if you choose Scary Monster you'd play the game as normal, choose Very Scary Monster and a small Huffin Puffin waddles in, forcing you to start it over.
In Mario Party 8, the team name for Mario and Dry Bones together is "BBQ Ribs".
The Reverse Minigame Bowser option in Mario Party 9, where the first person who loses wins.
Shy Guy's animation for getting third place in a mini game in 9. He does a little dance.
Yoshi seems to have taken up wagging his behind at the camera in some of his victory animations, which are also rather funny.
Toad getting 3rd place in 9. He shrugs and wiggles his hips.
In Mario Party 2's Horror Land, Mr. I's idea of teleporting you is... chasing you across the board.
The minigame "Apes of Wrath" from 7 has a beginning animation where the two duelists eat apples before being spotted by the Ukikis. If Dry Bones is one of the duelists, he bites into an apple... and his head sticks to it and comes off his body.
In Mario Party 5 and 6, if you land on a Mr. Blizzard space while carrying no capsules/orbs, a giant snowman falls onto your head while a text box reads "Nothing happened".
The Thwomp Orb in 5 is similarly hilarious if only for how abrupt it is. The character barely even gets to step onto the space before WHAM, Thwomp comes barreling out of the sky with no warning and crushes them flat!
In Mario Party 4, most of the Bowser minigame dialogue. A highlight is in "Darts of Doom", when Bowser forces the Koopa Kids to move the dartboard so that he won't get hit.
From the same minigame:
Koopa Kid #1: The player with the least points...
Koopa Kid #2: Gets a back rub!
Koopa Kid #1: NO! They get ROASTED!
There's also the way he sometimes in "Fruits of Doom" will hesitate before eating a fruit, as if wondering "wait, did I ask for this?"
Mario Party 7's Bowser Time had a couple funny moments:
The constant event where Bowser takes a memento of the trip and gets every player in a cut-out picture. The photo of the characters itself is usually normal for the most part... but sometimes Bowser "accidentally stumbles" into the photo. In the end, Bowser makes everyone pay 10-20 coins and dupes them.
Him making Koopa Kid rob a bank in Windmillville. On some occasions, Koopa Kid will fail to steal any coins, resulting in Bowser grounding his minion.
Any time Bowser mentions his bubble powers in Island Tour.