Do Nintendo DS cartridge batteries (do those games have batteries?) run dry after some time like GBA/GBC/GB games? Is it true that they will last at least 100 years or something?
A DS cartridge is like a micro SD card, so they don't have any battery inside.
And regarding life expectancy of an SD card I'll quote SuperUser:
Flash memory indeed has limited write cycles. However, by now it is unlikely that you'll encounter this within the normal lifetime of such a card. Usually this is in the order of 100,000 write cycles today and SD cards include circuitry to manage wear-leveling, that is, spread out writes over the storage media evenly to avoid "hot spots"—pages that are written too frequently and therefore failing early.
As most games save less than once or twice per ten minutes, I'd say that the electronics in a cartridge will go bad before you cannot save anymore. You will not be able to exhaust its write cycles. So, it should work for at least 40 or 50 years.