Bubble Bobble

Wonderboy III

Both I sometimes sing to myself when I'm sure I'm alone in a myriad of doododododo's.

Bloodrayne: Betrayal Official Soundtrack

Start listening from track 18 (Dusk Falls (NES)) for some great 8-bit lovey-dovey sensations.

In terms of 16-bit music, can't go wrong with the Streets of Rage music:
http://www.youtube.com/watch?v=iNO1Dlj087A
http://www.youtube.com/watch?v=CALdE0teZTs
http://www.youtube.com/watch?v=3hu5vCoxD_o
http://www.youtube.com/watch?v=2uY2_JPlKag
http://www.youtube.com/watch?v=7fA_wq_5ctM
http://www.youtube.com/watch?v=41eMgHquT9I
http://www.youtube.com/watch?v=olyldz02BxQ
http://www.youtube.com/watch?v=ufKwLdKMZek
http://www.youtube.com/watch?v=jo0wbrdFpSE
http://www.youtube.com/watch?v=KVRD50vXe0I
http://www.youtube.com/watch?v=_dql_FuYhaE

Had to limit myself since I love pretty much the whole soundtracks of the first and second game.

I've always really liked Metal Gear 2's music.

This track, this track, and The Theme Of Solid Snake are among my favorites. Snatcher has some really catchy tunes, but it's impossible to find the MSX soundtrack anywhere, and I don't think the Sega CD version counts as 8 or 16 bit.

My three favourite bits of Commodore 64 music

Thrust
http://www.youtube.com/watch?v=AQkf_H8zejU
http://www.lemon64.com/?mainurl=http%3A//www.lemon64.com/games/details.php%3FID%3D2641%26coverID%3D1604

Wizardry (the Youtube link is a bit odd, seems to be a remix of the game music and has a screenshot of the wrong game)
http://www.youtube.com/watch?v=8P6tvQoWKps
http://www.lemon64.com/?mainurl=http%3A//www.lemon64.com/games/details.php%3FID%3D2878%26coverID%3D1844

Gyroscope
The only youtube link seemed to not have much music
http://www.lemon64.com/?mainurl=http%3A//www.lemon64.com/games/details.php%3FID%3D3063%26coverID%3D681

Gyromite

http://www.youtube.com/watch?v=EGVTuXlsKgM

Last Ninja(c64)
http://www.youtube.com/watch?v=QOmin4vmMts

Human race(c64)
http://www.youtube.com/watch?v=zw2dqAxSo3I

Monty on the run(c64)
http://www.youtube.com/watch?v=4EcgruWlXnQ

Nostalgia systems engage. Firing The First Samurai:
http://www.youtube.com/watch?v=NfKC9EQ_WqI

Second salvo ready. Firing Gods:
http://www.youtube.com/watch?v=UnIlSgBrbRM

Target still standing. Finish 'em off with some Lost Patrol:
http://www.youtube.com/watch?v=ZffXCFTDY_c

Level 1 from the Choujin Sentai Jetman game on Famicom.

From Ganbare Goemon 3 on Super Famicom

And I think I'd be nuts to leave out "Daddy Mulk"

Original chiptunes or remakes of newer songs?

For original there is Journey to Silius - Stage 1:

http://www.youtube.com/watch?v=NjG2ZjPqzzE

For remakes I like this version of Persona 4 - Reach Out to the Truth

http://www.youtube.com/watch?v=c6jjCLfMWa4

Those are some of my favourites too, with Monty on the Run being my absolute favourite.

For fans of C64 remixes, I'll recommend SLAYRadio, a web radio channel exclusively devoted to the genre.

This is an absolute classic title from Rob Hubbard on the C64:
Master of Magic - Commodore 64

Star Fox Corneria

http://www.youtube.com/watch?v=eclAC0JQJYk

Castlevania Bloody Tears

8-bit

http://www.youtube.com/watch?v=EjazC45Qkww

16-bit

http://www.youtube.com/watch?v=yoQex8CB9eA