I'm looking for a flashcart that could play DS (obviously), GBA, and SNES/NES games. Are they any that could do this?
As suggested above, Supercard DSTwo is your best option, but IMHO it's not a satisfactory one. For what you're wanting, nothing is. I mean, the DSTwo is great, really great for DS games. Get a DSTwo for your DS games, you'll love it. The emulators that will run GBA, SNES, and NES roms on the DSTwo though .... well, it's mostly not bad, but I'm picky. I prefer playing GBA games on a GBA (or a DS Phat, or a DS Lite, or a Gameboy Player). I prefer SNES games on a real SNES or Super Famicom. And NES/Famicom is best on ... you guessed it.
If you have some real GBA hardware (including DSPhat or DSLite, of course), get an EZ Flash IV.
If you have an NES or Famicom, get an Everdrive N8 or a Powerpak.
If you have an SNES or Super Famicom, get an SD2SNES or Super Everdrive or SNES Powerpak. (google, compare, choose)
Until you have the original hardware though, I'd suggest using Nintendo VC, emulators on PC, or emulators on a hacked Wii/WiiU for the NES/SNES/GBA games.
Sorry to be a grumpy buzzkill.