You've got a distinction here that you may want to make first: NAND based or NOR based. Carts that use NOR flash are slow to write to, have limited capacity, and are very expensive. In addition, you cannot delete entries off the cart - you have to reflash the entire cartridge if you want to make changes (though you can add more games to the end without reflashing if you have room). However, they function almost exactly like an actual GBA cartridge once flashed. NAND based carts use either built-in flash or removable storage like CF or SD cards, which gives you a lot more space for less money. However, they usually have a few features missing, like the real-time clock and advanced savegame management. I personally have both, because I bought the NOR-based one several years ago for the GBA and then bought a NAND-based one later for advanced usage on the DS.
Popular NOR-based cartridges are the Flash2Advance Ultra and the EZ-Flash line, among others. They are expensive though. A 1 Gigabit (128 Megabyte) F2AU cart cost me $125, though it's at about $100 now. At the time, I was sorely tempted by the $70 XROM, which was 512 Mbits (64 Mbytes), as it was a good deal. However, for GBA gaming, it serves its purpose very well. These carts usually connect to your computer via a linker, with one end with USB and the other end with a serial port that plugs into your GBA's link port. These linkers usually also allow you to dump official games (I think I'm allowed to mention that).
NAND-based media cards like CF and SD are too slow for GBA gaming by design, so NAND-based adapters copy the games into fast RAM and execute them from there. While this shortens the battery life, it gives you the advantage of using a CF or SD card to begin with, which is much more convenient, as you can freely add, delete, edit, and move files to and from the card using a card reader. This means you can't dump your own games though, but you can purchase a linker cable on the side for $15 or so if you want to do that.
Popular NAND-based devices are the Supercard and M3, but since you're going to be using it for GBA gaming mostly, I'd suggest the M3, as the Supercard has slower RAM that makes some games slow down or simply crash. Get the M3 Perfect, not Pro (the Pro has very little RAM and therefore supports only very small GBA binaries), and I'd suggest the miniSD version (not the same as Slim SD) if you don't want it to stick out of the GBA. While the M3 is fairly expensive, you end up paying much less for the same amount of storage, and you can always increase your storage capacity by buying another cheap miniSD card rather than buying a whole new $100 cart like you would have to with a NOR-based cart.
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