Most of the games in the first run of White Wolf's Old World of Darkness setting were gigantic exercises in Shooting the Shaggy Dog. The good guys in each setting were gradually (or abruptly in Hunter: The Reckoning) revealed to have a long and unpleasant past of doing rather nasty things in the name of the cause, and the series of epic centuries-long secret wars they were fighting generally tended to be either unwinnable stalemates or tragically doomed noble causes. And most of their problems turned out to be caused by the arrogant hubris or ignorance of their own predecessors anyway. And to top it all off, the entire original setting had a series of apocalyptic end of the world scenarios as its grand finale.
Paranoia. It's essentially a game that states, in the manual, that every game should be a (funny) shoot the shaggy dog story.
In Call of Cthulhu, mythos monsters (and more mundane horrible experiences) make you lose "sanity points" and you gradually go insane. You get back Sanity Points by defeating monsters, which often require either magic or better weapons than the players have to kill. However, spells also cost Sanity, and most spells cost large amounts. If you don't go insane, it's all right, because most monsters can kill you anyways. However, it is justified by the fact that it is based on Lovecraft'sequally bleak books of the Cthulhu Mythos.
Similarly, the Trail of Cthulhu Purist scenario "The Final Revelation". The player characters must piece together a series of clues that suggest a vague threat to all mankind. In the very end, the characters find out that the horrible and monstrous alien gods they were trying to stop had actually devoured the world long before they even started their investigation. Their minds are totally shattered as the world dissolves into a surreal nightmare from which there is no escape. Talk about a Downer Ending.
KULT, a Swedish roleplaying game with its gloves off, the setting of which can be summed up as "Splatter Punk, Cosmic Horror Story, Mind Screw and lots of Squick". In KULT you can sort of "win", and become an Eldritch Abomination that you were before the Demiurge trapped you and the rest of humanity. Of course at that point there isn't exactly much left of you as you were as a human.
Don't Rest Your Head is made of this Trope: literally every game mechanic represents a different gradual (or rapid) slide of your resources dwindling away, as your life becomes more and more a nightmare. With a little luck and a kind GM, you may manage to save whatever is most important to you as you are destroyed in the process. If not... then this trope.
Fiasco: It is far more likely a character, for all their efforts, will end up with Shoot the Shaggy Dog ending than a good one. About two-thirds of the Aftermath Table are dedicated to endings such as "Pathetic," "Savage," "Grim," "Merciless," and "The worst thing in the universe." Given that the game is strongly based on The Coen Brothers' films, it's true to form.