Does the DS have a programmable sound generator?

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If so, then how many games make use of it?

 

[FAST6191]

I thought you would be asking for some kind of homebrew, for the sake of future forum searchers then https://www.gamebrew.org/wiki/Category:Music_homebrews_on_DS is a good start.

Yes, hardware docs
http://problemkaputt.de/gbatek.htm#dssound
and while they are for the GBA I also recommend
http://belogic.com/gba/index.php and http://www.coranac.com/tonc/text/sndsqr.htm

Most DS games will be using the SDAT sound format, and most that don't will be covered by https://gbatemp.net/threads/the-various-audio-formats-of-the-ds.305167/ . The various parts of the SDAT format (has a wave stream option, a midi like effort with samples and a simpler wave sample thing that tops out around 10 seconds but it is usually a lot less) have some looping capabilities but that is about as exotic as that gets (16 channels of arbitrary PCM sound and enough space that you don't really care too much tending to make for whatever you want without having to get too creative*).
If I were to look for something there then I would probably start with the music homebrews and whatever is in the commercial world (korg, electroplankton and whatever else). I don't think we have seen much in the way of system pushing for audio like we see in the NSF custom compositions world on the NES, though I have not looked for a while https://www.pouet.net/prodlist.php?platform[]=Nintendo+DS&page=1

*again for others then amiga and how it bypassed limits for some good stuff there