Courtesy of Loafmeister (Fellow Amiga Fan!:)

I believe the following are computer/console games (ie: not arcade conversions but I could be wrong on a few here) that were great
Amiga games. I've sorted them by category but some of these games can fit in multiple categories (Another World is an action game or
adventure (it's very cinematic) but I think it's also a platform game at heart)

Action: Turrican 1 & 2, Chaos Engine, Canon Fodder, Sword of Sodan, Leander, Alien Breed & Alien Breed Tower Assault, Batman the movie &
Typhoon Thompson and the search for the sea child

Adventure: Secret of Monkey Island 1 & 2 (though PC version is probably better), It Came from the Desert + Antheads expansion,
Rocket Ranger, King of Chicago, SDI, All the Sierra Adventures (King's Quest, Leisure Suit Larry, Space Quests, Quest for Glory,
Police Quests, etc), Rise of the Dragon, Willy Beamish, Heart of China, Indiana Jones Fate of the Atlantis, Beneath a Steel Sky, 
Cruise for a corps, Future Wars (might have been known under a different name, can't remember) & Loom

Fighting: IK+, Body Blows & Body Blows Galactic

Flight Simulation (don't think you need the full manual to play these): Wings (fantastic!), F/A 18 Interceptor & Battle of
Britain: Their Finest Hour

Pinball: Pinball Dreams, Pinball Fantasies & Pinball Illusions

Platform: Superfrog, Another World / (Out of this World - USA version), Prince of Persia, Shadow of the Beast 1, 2 & 3, Zool, 
James Pond, Robocod: james Pond 2, The Killing Game Show, Alcatraz (sequel to Hostages I think?), Gods, Stormlord, Rick 
Dangerous 1 & 2 & Switch Blade

Racing: Super Cars 1 & 2, Nitro, Lotus Esprit Turbo, Lotus 2 (this one is just awesome!), Stunt Car Racer, Formula One Grand Prix
(though the last PC version is the best and the Amiga version would be HIGHLY recommended... if you can enable acceleration board
support), Indianapolis 500 & Lamborghini American Challenge / Crazy Cards 3

RPG: Faery Tale Adventures, Dungeon Master & Chaos Strikes Back, Black Crypt & Eye of the Beholder 1 & 2

Shoot-em up: SWIV (pseudo sequel to Silkworm arcade shooter), Battle Squadron, Banshee according to google A1200 only however I'd 
have sworn I played this on my Amiga 2000... weird), Stardust, Apidya, Xenon 1 & 2 & Apocalypse (think choplifter)

Sports: Speedball 2 Brutal Deluxe, TV sports Football, TV Sports Boxing, 4D Sports Boxing (especially on an accelerated Amiga), TV 
Sports Basketball, TV Sports Baseball (AKA Bo Jackson Baseball), Archer Maclean's Pool, Zany Golf, PGA Golf, Sensible Soccer &
Sensible World of Soccer, Pro Tennis Tour 2

Strategy: Lemmings 1 & 2, Defender of the Crown, Shanghai, Bubble Ghost, Worms, Dune (or was it Dune2??), Populous, Powermonger,
Sim City, Puzznic, Pipemania (Night Shift USA version),

Misc (unsure where to place these): Nebulus, Virus, Laser Squad, Midwinter 1 & 2 (hard to get into but once you do...), Starglider 2,
Flood, Infestation, Escape from Colditz, Total Eclipse & Bombuzal

Note that there are a LOT more games, that's sort of just at the top of my head and a quick google search. There are also some amazing
arcade conversion such as Rainbow Islands, Pang, Super Hang On, Buggy Boy (Speed Buggy) Pacmania, Arkanoid 1 & 2, Silkworm, Ivan Stewart's 
Super Offroad, New Zealand Story, etc that can be used if Arcade emulation isn't available.

Given the time, I figure I could come up with a 500 games that are awesome to play. Yeah, there might be a few of those that have better 
versions on other systems but still, the Amiga was home to so many classics