Well I was looking at some of my games and some of them of 5 screws. I heard they are a little rare but I am not sure. Does anyone have a list of the 5 screw games. Also my other question is how many NES games save. Is there a list of them also.
Older cartridges were made that game 5 screws. They are not "rare". Most people don't collect variants but some do. Some variants might be considered rare or uncommon. But others aren't rare at all. There is a list somewhere of games known to come in 5 screw carts and ones that come in both the older and newer carts. There is probably a list of all sorts of known variants. Around 60 US NES games released have a battery backup. Here is a list that should be pretty complete. It's a relatively small portion of the NES library. Al Unser Jr. Turbo Racing Bandit Kings of Ancient China Bard's Tale, The: Tales of the Unknown Base Wars: Cyber Stadium Series Baseball Simulator 1.000 Baseball Stars Baseball Stars II Capcom's Gold Medal Challenge '92 Crystalis Déjà Vu Destiny of an Emperor Dragon Warrior Dragon Warrior II Dragon Warrior III Dragon Warrior IV Dungeon Magic: Sword of the Elements Faria Final Fantasy Final Fantasy II (Prototype) Formula One: Built to Win Gemfire Genghis Khan Greg Norman's Golf Power Heroes of the Lance, Advanced Dungeons & Dragons Hillsfar, Advanced Dungeons & Dragons Kirby's Adventure L'Empereur Legend of the Ghost Lion Legend of Zelda, The Magician Maniac Mansion Might and Magic: Secret of the Inner Sanctum NES Open Tournament Golf Nobunaga's Ambition Nobunaga's Ambition II Overlord Pirates! Pool of Radiance, Advanced Dungeons & Dragons RacerMate Challenge II Romance of the Three Kingdoms Romance of the Three Kingdoms II Shadowgate Shingen The Ruler StarTropics Tecmo NBA Basketball Tecmo Super Bowl Ultima: Exodus Ultima: Quest of the Avatar Ultima: Warriors of Destiny Uncharted Waters Uninvited Wario's Woods Where in Time is Carmen Sandiego? (Prototype) Wizardry: Knight of Diamonds Wizardry: Proving Grounds of the Mad Overlord Zelda II: The Adventure of Link Zoda's Revenge: StarTropics II
Probably to keep the costs down. SMB3 was expensive at the time due to the special MMC ASIC Nintendo added to the cartridge. The MMC was used to enhance the power of the NES and provide extra graphical functions in the game, like the split screen effect and so on.
If you want Super Mario 3 with a save feature, get Super Mario All-Stars on SNES. As Taucias said, cost. If you have a flash cartridge, someone I believe went to the trouble of making a hack that allows you to save your data.