Wizkid: The Story of Wizball 2:http://www.youtube.com/watch?v=V12cYRmlekE
Goofy, weird game that is kind of uncategorisable. You switch between a block-bouncing arcade game where you destroy enemies by dislodging objects and knocking them into enemies, and a 2d platformer adventure with dream logic puzzles. Genuinely great game.
Driving/light gun shooter hybrid. Fun, original arcade game that went beyond the often-dull shooting gallery that lightgun games usually are and gave you more control. A prototypical FPS.
Inspired by Thexder (released a year earlier), a 1986 action RPG that predates Zelda 2 and its pals. You control a mech that can transform into a tank and a jet. Run, drive and fly around a very non-linear 2D world map with enemies wandering around. Run into them and the game switches to a 2D realtime combat view. Enter buildings and you move around 3D polygonal dungeons - with an automap (rare for '86). Upgrade your mech with max enery/ammo units, collect keys, get XP - fun times.
The first sidescrolling platformer, released in arcades a few months after Donkey Kong. Collect money bags for an extra life (and points thereafter) and enter a multi-directional scrolling pyramid. Actually a fun game.
First game with filled polygons and shading, but so much more than that. Not really categorisable, but a kind of hybrid platformer/shooter/maze game. Allows you to switch camera angles (and gives you more points if you have it at a lower angle). Tons of great, strange ideas.
Scrolling shooter from the people who did R-Type but way less rigid and memorisation-based - game gives you the tools you need to survive, but it really makes you use them.
Fun, original platformer with novel, interesting gameplay.
Cool, weird arcade shooter released in '85. Save strange little one-eyed freaks and at the end of each stage each one increases your chance of winning the roulette wheel minigame, thereby increasing your max health. If you get to a boss and you still have some, they come out and suicide-attack the boss, stunning it and allowing you to get a few free hits in. The main character, Kissy, is actually a girl as revealed at the end of the game, beating Samus Aran as the original striptease heroine. Kissy was married, and divorced, from Dig-Dug - their kid is Mr. Driller.
A fps in 1985. Awesome fractally-generated cave walls that still look cool and interesting gameplay. You shoot enemies with shots of the same colour to defeat them, or polymorph them into something of a different colour if you don't have the right kind of ammo.
I will add more when I think of some.