Everyone knows The Legend of Zelda was the first NES game with a save battery. However, the Famicom version doesn't have one, since it was originally a Disk System exclusive (until a 1993 re-release, anyway.) So, what was the first Famicom cartridge that had one? The Japanese versions of the first two Dragon Quests use password systems, so it's not them.
The earliest Famicom cartridge I'm aware of with a save battery is Mirai Shinwa Jarvas, released June 30 1987 (which, incidentally precedes Zelda's US release that August.) Does anyone know if there's anything that precedes that, before I go and determine the answer myself?
Morita Shougi precedes that by a few months and allows for saving a game in progress. There's only three things earlier than that which have any battery at all and they're all BASIC programming environments.