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RiverNotch would that game involving solving a variety of puzzles on an island be Island of Dr. Brain, by any chance? It certainly fits the description.
The entire Ace Combat series has good arcade flight sim gameplay and excellent music, not just Zero.
I don't know what my favorite retro game is (or where we're drawing the line), so I'll just put down the entire old scrolling shoot-em-up genre. All of it. Love those games, even if I'm terrible at them. I'm particularly partial to R-Type...as one might be able to guess. Final's the only installment I've really played thoroughly, but the games are all fun. Nobody makes any games like them any more, except for Japanese indy developers, it seems.
Also, going to mention Metal Slug 3 as a great game. Not exactly the same genre, but definitely the highlight of the Metal Slug series. I could keep listing games that I keep coming back to or particularly love, but that's not quite the point of this discussion.
Also, just to comment a bit on the first post, Parasite Eve is a great game, and the first is definitely the best. Nomura did some really great monster designs for it, and I kind of wish he had more opportunities to do that sort of stuff.