So, what weirdnesses have you loved? What games (or settings) were so memorable that you still think about them constantly? When you think of RPGs that have really pushed the bounds of what is even possible, what games do you want to talk about?
Weirdest:
1: Brute Squad: a rules light difficulty budget game, which has a claim to being fortune first. You state the Brutistic (attribute), roll it, then describe the resulting effect. If the table things your description doesn't work for the roll, complications happen, and the current obstacle is failed.
2: Blood & Honor and Houses of the Blooded (John Wick): you don't roll for success, you roll for who decides and in what order you get to "Yes, and..."/"Yes, but..."; the number of dice omitted from your pool determines how many YA/YB statements you get. In combat, initiative is in order of high roll to low roll, but number of actions is by omitted dice. B&H I've run, it's a samurai pillow-book theme.
3: 7th Sea 2e (John Wick) - At start of a conflict scene, you roll for number of actions in the conflict; scene ends when everyone's out of actions. That's where I quit reading. (I was having to copy and paste to be able to read the text; John's font choice makes HoL look easily read.
My favorite settings
for play aren't that weird... L5R (Esp 5e), Alien, Talisman Adventures, Star Trek, Space 1889, Star Wars.
I still love Traveller's Third Imperium (at least the non-mongoose versions), but for play? My player base no longer is able to really grasp the tropes well.
I want to run a Mecha Anime game (other than Palladium's Robotech), but my players in both groups aren't fans of the genre. Both Heavy Gear and Jovian Chronicles have good settings, and Mekton has Mekton Empires... but those really aren't weird, either.