I'm intentionally ignoring the generally agreed-upon games, such as Megaman and Castlevania, and concentrating more on lesser-known games. Some of these are of a particular song, while others are of an entire game's score. There are loads of remixes of some of these, and none for others. I posted a few of these where applicable. Also, these are in no particular order, because I don't feel like sussing out which I like more. Oh, and as per the nature of chiptunes, they're pretty short and repetitive. Forgive them.
1. Abadox - Stage One Theme:
I borrowed this game from a friend when I was young, and I was immediately hit by the awesome first song. Unfortunately, the rest of the game's music is kind of bland, and the game is nearly impossible. I want a remix really bad, but can't find one.
2. Jackal:
This is like NES chipset power metal. A classic game, and the entire soundtrack is sick.
3. Astyanax - Remlia:
Another in the "shitty game, good music" category. Remlia has one of the best themes I've ever heard in a game. The remix I posted below is also incredible.
4. Ironsword - Title Theme:
I had this game when I was a lad, and I loved this song so much.
5. Journey to Silius:
Not a well known game at all. The entire soundtrack was so good. Like Castlevania 4 on the SNES, the drums and bass were sampled. Add that to the great composition, and its a memorable soundtrack. There are some ok remixes out there, but I want a really well-done mix of the stage 3 song. Its so good.
6. Goonies 2 - Good Enough:
Of course, I'm cheating a bit again, but this digitized version of the Goonies theme is just so well done. I think I like it more than the actual song.
7. Double Dragon 2 - Forest:
Simple, but catchy. There is a whole album of official remixes of this game.
8. Solar Jetman:
Another childhood memory, the songs are all short, but they're very good.
9. Ys - First Steps Towards War:
More well known than the rest on this list, Ys consistently has excellent music. There are so many arrangement and remix albums of these, you're sure to find them if you search. Oddly enough, I can't find the actual NES chiptune of this song, which is my favorite from the game.
10. Batman:
The rare licensed game that was actually fun. The music is properly dark.