With the exception of Pinball Illusions, every game suggested so far has been an Amiga 500 game. Since you just bought an AGA Amiga, I thought I'd suggest some games a bit newer and more visually impressive.
Give these a go:
For a stock-standard A1200:
Fightin' Spirit AGA - Arcade Fighting Game
Super Stardust - Raytraced Asteroids Remake
Slam Tilt - The Pinball Simulator
Aladdin - Platform game based on the Disney film
Banshee - Arcade Style Aeroplane Shoot-em-up
Kargon - First Person Perspective Fantasy RPG
Virocop - Destroy Viruses in an Infected Amiga
Overkill - Deluxe Update of Defender
Shadow Fighter AGA - One-on-one Beat-em-up
Speris Legacy - Cute Adventure RPG
Roketz - Fantastic 2-Player Gravity Shooter
Simon the Sorcerer - Funny Fantasy Adventure
Skeleton Krew - Console-style Gritty Blaster
Blitz Bombers - Multiplayer Bomerman Game
Worms: The Director's Cut - Comedic Strategy Wargame
Nick Faldo's Golf - Easy to play Golf
Deluxe Galaga AGA - The Ultimate Remake
Lords of the Realm - Medieval Trading and Strategy
Max Rally - Multiplayer Overhead Racing Game
For an expanded A1200: (Sorry, too tired for descriptions)
Genetic Species - 3D FPS
OnEscapee - Adventure/Platformer
Land of Genesis - Platformer/Shooter
Payback - Driving
T-Zer0 - Shooter
Virtual Ball Fighters - Puzzle
XTreme Racing - Racing
Wasted Dreams - Adventure
Bubble Heroes - Puzzle
Capital Punishment - Fighting
Final Odyssey - Adventure
Foundation: Gold - Strategy
Napalm - Strategy
