I'm going to keep this one strictly on D&D just to narrow it down some. I've played so many games with so many systems with hilarious events that I can't possibly pick the "craziest thing" out of all of them, especially when Paranoia is on the table. So D&D only. Here we go.
The first one I can think of is when a Paladin (hey Wes!) in a new game I was running mistook a Cockatrice for a Basilisk. What's the big difference, you ask? Well, I already wrote about the hilarious results on my old blog back in the day, so I won't retread old ground. You can check it out here. Serious lulz.
I guess I'm going to go with a game I was running where the players were running an espionage/sabotage mission, each in separate area of a "dungeon" (they were actually inside a giant mechanical golem, but...nevermind that), and a player in my game, Valgyar the Monk, who was invisible, flying, and Bull Strength'd, decided he'd sneak/float up to a Minotaur that was standing at the edge of a ledge and push him off in a surprise attack. He could've just snuck by him, but the temptation was too great. I added up the ridiculous bonuses he had and realized it was like a +16 for a DC 15 attempt (3e rules). So I smirkingly said: "Sure, roll anything but a 1." (on a d20) However, this is the same player who played Marcass (the Carcass) all those years ago. Naturally, he rolled a 1. The Minotaur snatched him out of the air (canceling his invisibility) and beat the hell out of him. Oh, and all the alarms went off, alerting all the guardian monsters and ruining the sabotage attempt, and the other players had to rush through traps to get to him before he died (again). It was a ridiculous (and incredibly unlikely) failure that led to the entire carefully-crafted sabotage plan by the players to end up being a 3-hour straight up survival brawl (and near TPK) inside a giant metal golem/dungeon.