Madness is an aspect of life, and so something that might be simulated in a roleplaying game. The venerable Advanced Dungeons & Dragons had rules for insanity, on pages 83-84 of the original Dungeon Masters Guide. It listed 20 different types of insanity that could be caused by various magical afflictions, curses, or psionic attacks. Among them was a type of "lunacy" which literally linked attacks of madness to the phases of the moon, paying more than reality's mere lip service to the derivation of the word.

Probably the game that best utilises madness, though, is Call of Cthulhu. Any PC who doesn't eventually go mad in that game is highly unusual, and very likely doing something wrong.

Much fun can be had by combining the sanity rules of Call of Cthulhu with other games. Realistically, the sorts of things that happen to adventurers are pretty stressful. Attacked suddenly by orcs? Make a sanity roll. Strange iron statue comes to life when you try to pry the jewel out of its eye socket? Make a sanity roll. Ambushed by bandits? Make a sanity roll. Your horse throws a shoe? Make a sanity roll.

Okay, maybe not.

Transcript

Anakin: Give me control of the military. I need their power to hunt down the Sith who have infiltrated the Jedi Order.
Palpatine: No; absolutely not.
Anakin: I see the time has come for more than gentle persuasion.
[SFX]: Whooom {Anakin draws his laser sword}
Palpatine: Are you threatening me?
Anakin: Call it what you will.
Palpatine: Strike me down then; you won't get what you want.
Palpatine: There must be another way; a peaceful way.
Anakin: The Sith are beyond peace. Don't you understand! Strength is the only thing they respect!
Palpatine: I have the strength of my convictions; that shall have to be enough.
Palpatine: No, Anakin, you must find another way; I will not have the Republic plunged deeper into conflict.
Anakin: Very well. I have always respected your integrity.
Anakin: I will pursue the Sith as best I can. I pray that it will be enough. {leaves}
{beat}
Palpatine: No wonder the previous Chancellor went nuts.