Just thought I'd throw my few cents in here....
The most resilient card I've used is the DSTwo or DSTwo+*. It's true that most carts allow you to use Action Replay cheats and many have save states. The DSTwo cards offer a Free Cheat function where you can search for your own cheats, a very nice and fast Save State Manager, some modest file management systems, has survived round after round of Nintendo trying to kill it. The SuperCard team is really good about keeping the card's firmware updated to work with the latest n3ds firmware. There IS a bump on the cartridge that makes it a snug fit into the console, but I own both variations and there's no problem inserting them. Sometimes you do have to wiggle it to get it out.
That being said, a few notes about the differences between the 2 cards and a warning:
The DSTwo and DSTwo+ are essentially the same exact device in a sense that the 3ds & the "new" 3ds are the same device. They both do the exact same thing except the Plus model has better hardware that can (by use of a special plugin) run 3DS roms using Gateway's launcher software. DO NOT USE THE 3DS FUNCTIONS OF THE DSTWO+!!!! If you set it up wrong or use the wrong plugin/launcher combination, you can end up with a hard bricked console....End of Warning!
Last time I checked, the DSTwo+ was actually cheaper than the regular DSTwo (likely because the SuperCard team can't seem to fix the 3DS issues). The Plus does seem to run emulated games a bit faster, if that interests you.
If you just want to run DS roms and you don't care about features, any el cheapo cart will do (assuming it's compatible with the firmware you're on). I had pretty good results running the r4 3ds rts cart, which I believe costs around $10-$15 through most vendors. This cart does work with Action Replay codes and has some Real-Time Save and Slow-Motion features, but I never used them, so can't say how well they work.