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I actually enjoy most of the games with this type of difficulty. I tend to find myself sporting a killgrin whenever I'm on a roll and cheating death every which way. The "OHSHITSHITSHITSHITSHIT" factor adds a lot to the rush.
However, video game logic gets especially offensive here. Getting grazed by a stray bullet. Collision damage, etc. C'mon, man.
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You can make some ballin' Demon's Souls and Dark Souls characters if you're willing to play under this principle. Not putting any points into your HP means you can put those points elsewhere, opening up a lot of interesting build options.
(Dark Souls even has a bullet hell boss fight, which makes this kind of character even more appropriate.)
I'd like to see a beat em up try and pull this off. The only game I've played that fits that description is My Hero/ Seishun Scandal, some old Sega arcade game. It was... kind of blah.
If I were to design a single hit-point beat 'em up, I'd make sure there's a decent parry/ rock paper scissors system ala Bushido Blade 2, or the Cross Counter attacks from Punch Out!!
Some of the later Arcana Heart bosses might as well be this.
Oh Gawd, the cruelty. It BURNS!
Seriously though, this would be rather interesting. The game would probably be ridiculously hard but anyone who finished it would probably get extreme feelings of satisfaction and badassery (both of which being justified).
I think there would be ways to do it with normal difficulty if the terrain were well-designed and the combat revolved around crowd control.
Most arcade-era beat-em-ups might as well have been one hit kill. Wouldn't have changed much.
I mean they were designed with the explicit purpose of being pretty much mechanically impossible to win without eating a shitload of coins.
Ninja Combat is pretty much that after the first level and SPOILER it's awful.
Violent Storm, Undercover Cops (the non-butchered versions, that is), Metamorphic Force (again, non-butchered), Double Dragon, even Final Fight etc. are very possible to clear on one-credit or less. There's this wonderful youtube channel full of legit brawler 1CC clears.
Dante Must Die Mode is OHPW, in Devil May Cry. But then, in that mode so is everything in the game.