NES

There's alot. Any game that has a battery back up has a save feature. Most RPGs had batteries, although some used passwords instead (damn you Swords & Serpents!). In addition, alot of later titles from other genres had batteries, including Kirby's Adventure, Formula 1 Built to Win, Capcom's Gold Medal Challenge, and a plethora of others. If you're looking for a list, it's gonna be at least 100 or so titles, way too many to actually list here.

There is a warning or notification on the back of some (most) games that will say there is a batery in it those will save. As a small pointer you can replace them with a watch battery if the batery is dead.

Most Nintendo clasics and capcom games are with the exception of like Megaman. Its mostly just a hit and miss if you arent looking at the stickers. I would recomend finding an emulator and then loading roms to see wich ones have save functions and then getting games based on that.

As my answer is incredibly long, it's still worth the 30 minutes.

As in NES, battery back-up is new and expensive at the same time. Here are the two versions of the NES (USA & PAL identical), each with their own way of saving.

Famicon & The Disk System: They use a battery back-up ability in order to save, in the expense of loading screens. Oh, and it's only in Japan (emulators now available). Games like Doki Doki Panic use this save.

As for our region of the NES, the only known way of saving is a built-in password. Every time you complete something, you are given a password (write at the ready) and you must type it to get back to your progress. Games like Gremlins and Joshua use this mode of saving.

As for the rest, sadly, you cannot save (NES-wise).

Which console (or generation in video gaming) are you talking about?

Retroreviewer he's asking about the NES as this is the NES board