For long-term playability, some of the classic Amiga games have really lasted for me. The multiplayer games last longest.
Here's a small selection, though I note that the Amiga, like the C64 before it, had silly amounts of incredibly good games. The ones I've listed here are ones I consider really long-lasting, there were many other excellent games such as Uridium II, Turrican II, X-Out, Cannon Fodder, Sensi Soccer and so on that were also very good, though.
PC owners, particularly American ones, often associate some games that I class as "Amiga" games with the PC rather than the Amiga - many european games started out on the Amiga, and were ported to the PC for the American market. The Amiga, at the time, had far superior 2D graphics hardware, with very smooth and fast scrolling, and a coprocessor (the "copper") that made split-screen gaming very easy. Thus, while some readers may associate some of the games with the PC, the Amiga versions tended to be much smoother and more colorful.
The one that really stands out in my mind is the original Amiga Lemmings, with it's simultaneous two player split-screen mode. You could plug two mice into an amiga at once. Each player got half the screen, one with blue lemmings with green hair, and the other with green lemmings with blue hair. On the same level, at the same time. Total chaos.
Worms, the Director's Cut:
Worms started out on the Amiga, as a Blitz Basic program by a lad in Yorkshire. The Directors' Cut is still, to me, the purest and best version of the game, particularly since you could draw your own levels and algorithmic level themes in DPaint or PPaint.
Again, really shines in simultaneous two-player. Play it to this day. I have a lovely logitech optical trackball that works very well with it.
Null-modem two-player god game. Rain down fire upon thine enemies. Great fun. Tricky (but apparently possible) to get two-player working under emulation.
Two-player vertical scrolling shooter, sequel to Silkworm. Action never stopped - the game was continuously spooled from disk. Knackering. I never completed it. Also adapted to your playing, with extra enemies activating if you were "doing a bit too well". Asymmetric - one player in a helicopter, the other in a bizarre jumping jeep. Meant that I could play in the jeep, while my little brother played in the easier helicopter. A later game called Banshee was quite similar.
Non two-player amiga games I'd still play:
The relatively unknown sequel to Wizball. I think it was cool, anyway. Too strange for some people.
The best Asteroids clone I've ever seen. Amazing graphics, for the time, and great gameplay. Unfortunately, used every dodgy hardware trick in the book, and doesn't work even on my accelerated real amiga, let alone an emulator.
Bob's Bad Day:
Another strange game. You're a head. Gravity pulls you to the bottom of the screen. You rotate the level around you, bouncing around the place. I think one of the Sonic games had a bonus stage a bit like it - but bob's bad day had a lot more levels...
Wizball on the C64 (the amiga version wasn't as good). Another asymmetric game - one player Wiz, the other player the much smaller cat. Very strange. Very cool.
Head-over-Heels on the C64 (The much later Amiga version is equally good)
Don't eat yellow snow