The original Streets of Rage had killer clowns. Apart from constantly running away from you and taunting you, they also attack in large number or with other minions to help them. They are always one inch too far for you to hit them, and if you do manage to hit them, combos are nigh impossible since they constantly produce torches or axes out of thin air (you have to punch all of those out his hands before you can attack him). They are even worse in Streets of Rage Remake, where the new AI makes them even more difficult to hit or grab.
The Signal dudes can often hit distracted or busy players with their sliding attacks and they can do throws too. Though you can land in your feet by holding up on the d-pad, watch out for Bottomless Pits...
The first Final Fight had Js and Two-Ps. Alone, they were pitiful, but if you turn your back on them, they rush in and nibble you. This gets much more dangerous if they are accompanied by tougher enemies - drop your guard for a second, let them hit you and the next thing you see is an Andore piledriving you for half of your lifebar before you had a chance to react.
In the N64 Rampage games there is a specific enemy that is basically a man in a blue jumpsuit with a jetpack and a flamethrower. They're hard to hit, take two punches to kill, and can knock you off buildings and temporarily paralyze you by setting you on fire.
Veiled ninja Shinigami in Bleach: Soul Carnival 2. Not only do they run faster than you can, they can also teleport away, leaving behind a treasure chest with a bomb inside it. If you hit the chest, you take damage and get knocked back.
This was the main problem with the Sega Genesis version of Mighty Morphin' Power Rangers: The Movie: too many mooks to kill at every stop of a stage. It gets to a point where it pretty much becomes Survival mode: you fending off wave after wave of Putty Patrollers (who are even worse in this respect than Ivan Ooze's Oozemen from the first two stages) as you hope that the game will let you go ahead.
The fourth level of Grief Syndrome is a seemingly endless gauntlet of snakes and floating seahorse-things. They only take damage if you hit their heads, which are way up off the ground at varying heights and quite small, so you'll frequently miss. Even if your aim is right, half the time they'll teleport behind you and you'll miss anyway. They attack in large, disorganized groups, and there's literally nothing else in the level besides dull gray terrain until you reach the boss. Oh, and you have to deal with them while fighting the boss, too.