When it comes to role-playing games I'm a collector. I accumulate piles and piles of books for different systems and settings and I hold on tight and never let go. Obviously this has led to vast piles of game books that I have never played or run (and sadly I may never get to). The upside is this gives me plenty of options to choose from when I'm looking for games to scratch a specific itch. Is the group looking for straightforward, kick-in-the-door dungeon crawling? I've got a couple of editions of dungeons and dragons to choose from. Do I want high action, risky, pulp-style gameplay? I pull out Savage Worlds. Super-heroes? PDQ's Truth and Justice is the best I've ever played. Epic, over-the-top insane action? I take my Exalted books, throw out all the rules and create a mad hodge-podge abomination out of several systems to try and get the right mix. I'm still working on that.
Sometimes I play games that are more "universal". Games where just about any character concept may be appropriate. I've gone through a few different go-to systems for these sort of games. The first attempt involved GURPS...but that's a horror story I'll save for later. You'll recognize it because the title of the post will probably be "The Worst Game I've Ever Run" and I will write it while weeping. However, I've noticed that no matter what system or setting I use the players will tend to go utterly insane when it comes to character design. I don't mean "I'm a werewolf cop, hanging out with a ninja, a pair of robot twins and a technowizard." I mean characters that look like the combination of a H.R. Giger painting and a random word generator.
Other than the Game That I Will Not Speak Of (yet) my first shot at a universal game was an adaptation of Rifts. For those who don't know Rifts is a futuristic, post-apocalyptic earth setting where magic has returned, portals have opened to pretty much everywhere and the world basically resembles a heavy-metal album cover. This sounds amazing but the system itself is actually terribly antiquated, unbalanced and generally clunky. So, I replace it with the current edition of Big Eyes Small Mouth. A relatively simple (compared to say GURPS) point-based universal system. By no means is BESM an incredible system either but it's faster and simpler than the core system in Rifts while allowing pretty much the same sort of characters.
Looking through the setting there are all sorts of crazy badass character concepts: newborn dragons, giant mech pilots, cyborgs, super-psychics, cyber-knights, etc. etc. However, my players choose not to go with any of those. Instead I get three of the most bizarre characters I've ever played with. The first is Hyako, a timid young man with some fairly impressive empathic and psychic abilities who (when stressed or angry) transforms into an amazonian female swordswoman capable of cutting robots in two with her katana. Next we had a small wizard in tattered robes. Normal so far right? Underneath those robots is a sentient hive mind composed of carnivorous moths that can form into a humanoid shape or break into a swarm to devour enemies like a piranha. Finally there was mega-man. And I'm not kidding. The player created a robot in the shape of a young boy equipped with a built in arm-cannon and a laser sword named Crescendo (keep in mind that the original mega-man characters all had music-themed names. like Rock, Roll, Bass, etc). Also, the 'not-mega-man' robot boy was perhaps the most angsty and emo character I've ever seen.
Another common setting I've seen this is games I've run set in Illuminati University. That's a GURPS setting that features a cross-dimensional university which is attended by wizards, mad scientists, super-humans, demons and other insanity. By this time I wasn't enthusiastic about the GURPS system but I do love the setting. So my standard solution is to use the PDQ super-hero system Truth and Justice to allow just about any character. So what sort of insanity do I end up with. One character is essentially an alien that resembles an animated, telekinetic windchime made from crystals. Another character is a Neko-mancer, a wizard who can perform a variety of magical spells so long as they relate to cats. Another character is a 2-foot tall, one-eyed green alien with a long, prehensile tongue and a stomach that is basically a giant Bag of Holding. Finally there was Maul...an 8 foot tall, morbidly obese humanoid embodiment of gluttony. Maul had a fair amount of super-human strength that only got stronger the more he ate and ate. He had a giant fanged mouth...plus two smaller mouths where his eyes should be...and a fourth mouth positioned on the back of his head behind his long black hair...which was prehensile and which he used to feed himself continuously from a backpack full of junk food. By the way, Maul was my girlfriend's character.