I haven't got much to add, unfortunately - everyone seems to have covered all of the great games, most of which are still fun today!
Don't think anyone has mentioned Skidmarks/Superskidmarks yet, nor the original "top down" Alien Breed games. How about Assassin (for that matter, mostly anything published by Team17!). Zool was also good for a laugh... International Karate + and Body Blows were pretty cool beat 'em ups (much better than the ports of StreetFighter2 and Mortal Kombat).
If you've got a powerful-enough Amiga (not an A600 I'm afraid!) then you can even run stuff like Doom, Quake, Quake 2, Descent, Freespace... pretty impressive, although dated - it's a shame that it took so long, that the Amiga scene could never catch up
As you've got an A600, there's a few that others have mentioned which won't run properly/satisfactorily - SimCity will be OK, but SimCity2000 was a dog even on a top-of-the-line Amiga. Frontier really needed the '020 processor in an A1200 in order to be fun. Formula 1 Grand Prix takes on a new lease of life when you use some of the performance patches that were available, etc.
If anyone really wants to relive the nostalgia, the best solution is to get an basic A1200 from eBay (dead cheap), buy and fit a cheap hard disk (4GB or less - v. important!), get hold of a piece of software called WHDLoad (which patches floppy games to run on your machine, directly from the HD - no more floppy swapping!)... grab your zipstick and have FUN!
Some sites worth checking out if you want a bit of nostalgic banter:
http://www.amigaworld.net http://www.amiga.org
and if you want some Amiga-specific classified ads:
http://www.amibench.org