Wizkid: The Story of Wizball 2:
http://www.youtube.com/watch?v=V12cYRmlekEGoofy, 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.
Night Stocker:
http://www.gamesdbase.com/game/arcade/night_stocker.aspxDriving/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.
Wibarm:
http://www.youtube.com/watch?v=kG-DqEN5vBAInspired 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.
Jump Bug:
http://www.youtube.com/watch?v=E_P--FYbhEUThe 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.
I, Robot:
http://youtu.be/y-7dT2c_bX4First 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.
X-Multiply:
http://www.youtube.com/watch?v=ikx26GYrKaoScrolling 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.
Spanky's Quest:
http://youtu.be/tqCgq9TvCc4Fun, original platformer with novel, interesting gameplay.
Baraduke:
http://www.youtube.com/watch?v=nG2XGUUDmJACool, 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.
The Eidolon:
http://youtu.be/E9cTvuwYi3wA 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.