Cloudcuckoolander/Tabletop Games

Card Games

Magic: The Gathering: the Ravnica block gives us the Izzet. Red/blue, it turns out, isn't quite right in the head, considering their hobbies include routing magic through the heads of goblins with an interesting variety of psychological conditions. (The goblins, that is, not the magic. Until the magic has gone through the goblin, that is.)

Foxbat from the Champions setting is completely convinced that he is a great, powerful comic book Super Villain.
Some of the Malkavians from Vampire: The Masquerade could get this way. They tended to be really scary at the same time if they're played well.
"Mad" Jack O'Bannon, king of Inismore, from 7th Sea.
House Criamon of Ars Magica. They view the physical world as a series of metaphors and symbols and spend their entire lives trying to interpret everything around them the way literary scholars interpret novels, so getting a Criamon to express her thoughts on anything is a little bit like communicating across a language barrier via Babelfish translations.
The Mystara D&D setting is home to the d'Ambreville family of wizards, who seem susceptible to an hereditary strain of this trope. As any d'Ambreville who isn't a Cloudcuckoolander is usually a freakin' trigger-happy psycho, siding with the loony ones is usually the best bet.