I think people aren't thinking this thoroughly or rather go for a selfish choice that would seriously put a huge dampen on the gaming community. We would be talking about not being able to play anything differently except for what the franchise has to offer, and that is something that you have to really think through when trying to reply to this, not just your favourite franchise.

I'll talk about my case. Final Fantasy. Do I love Final Fantasy? Yes. Do I think it's the greatest franchise ever? No. Why Final Fantasy, then?

· Because it has over thirty games (quantity, so that people, at least, get a lot of games to choose from, if not play them all since there's nothing else).

· Because the formula is changed in every entry (meaning gameplay will be, in its majority, fresh between each entry, and there's something different in each case).

· Because there's a lot of spin-offs games that cover a multitude of genres (Shooters -Dirge of Cerberus-, Strategy -Tactics, Revenant Wings-, Racing -Bikes-, Rythm -Theathrythm-, the whole Chocobo games and the Tower Defense games -Crystal Defenders-, loot-based adventure games -Ring of Fates- and social multiplayer ones -FFXIV, Crystal Chronicles).

· Because there's something in every platform conceivable, meaning people would had multiple ways to get into this - only - gaming franchise.

· Because the merchandise from the franchise allows people of experiencing it in multiple ways than one: books, comics, anime, movies, figurines and all other kind of physical and digital items.

Now let's examine some people's answers:

- OP said Battlefield. Cannot stress just how bad this would be. Sure, FPS enthusiasts and FPS multiplayer gamers would have their time of the day, but what about singleplayer people? (Their only relieve would be Bad Company 1 and 2, which feature a solid and enjoyable campaign). What about people who don't enjoy FPS games? At least there's diversity having games set in almost every possible scenario, but this would alienate so many players it's not even funny.

· The Legend of Zelda. Now this franchise has had a long tradition and features many games. Quantity is assured, but on diversity lies the problem. We have two different versions: the isometrical, generally 2D games, and the 3D ones; both follow the same structure in most cases (There goes Wind Waker and Spirit Tracks to Combo Break player's traditional progression xD), and the gameplay is generally the same as well. There's some small side adventures like Hyrule Warriors for the Dynasty fan, and...eh...Link's Crossbow Training for those who want to go back to the on-rails day. I can think of one social adventure game (Four Swords, there's that Tri-Link game on 3DS as well), but the rest would be purely exclusive to a singleplayer affair that is, at its core, unchanged in most entries. While certainly a better choice than Battlefield for singleplayer gamers, this franchise would really alienate a lot of players as well, so it's not that good of a choice.

· Fallout. It sure gives hundreds and hundreds of hours, and while 1 and 2 are radically different to 3 and 4 (meaning there's that fresh sensation to experience), quantity isn't assured in this case. There are some side projects like that strategy game where you're a Brotherhood member (can't remember the name, Tactics it could be?), but you'd be mostly left to those four mentioned games. Mod support can take 3 and 4 very far, but to be fair here, unless you truly enjoy the core mechanics offered there isn't much here for most players, and no way for social gaming unless some miraculously multiplayer mod like the one in Just Cause 2 is added, which isn't happening anytime soon.

· Metal Gear Solid. This franchise would simply left aside a huge percentage of the gaming population. While a gaming favourite franchise, with a moderate/huge success in most cases, it would be hard for most people to get into some espionage games on which you tackle down huge mechanical monsters after the President of the United States touches your crotch to confirm you're a man. Just saying. Not to mention replayability isn't all that great in most cases, although there's a fair percent of social gaming here counting the spin-offs.

· Souls/Bloodborne. Talk about the epitome of alienation.

I think Mario would be another great franchise that cover most bases, if lacking any kind of massive multiplayer online, but still a much better choice than many others.

Surprisingly, mine was not made from a selfish choice, haha. Mario isn't my favorite video game series, but I felt it is one I thought would do well in covering a large varying number of bases. There's single and multiplayer games, RPGs, sports games, party games, puzzle games, and racing games. It's missing a couple of genres, but I think it was a good fit to cover as much ground as possible.

Final Fantasy while may have some varying games, I still don't feel they appeal to everyone. Many of their games are focused on single player and RPGs, which I don't think every one would be into. And the games it has in other genres, they don't have as many in others as they do as Mario. The different genres Mario cover has at least a couple of games in each one. Plus, Final Fantasy has done some weird questionable things with its game.

In a way, Mario was designed to appeal to as large of an audience as possible. Sometimes that's their aim, like they did with New Super Mario Bros, which was aimed to bring people who fell out of gaming back into gaming, and that succeeded from what I recall from sales.

Anyways, that's my case for Mario.

I mentioned that Mario was indeed a nice, thoughtful option considering the premise of the OP. It has over fourty games, and there's an arrange of diverse genres and themes present on them. Regarding Final Fantasy not being appealing to everyone, that's true; but the fact that the games try reinventing themselves on each entry means that (considering you're pretty much forced to play those or nothing) there's gotta be at least one that you eventually like. Doesn't even have to be a mainline entry (but it's questionable, considering how much the gameplay has changed after all these years), you could just chill around with Crystal Bearers and have fun with that one instead. Mario, despite having entries on diverse genres, is consisted mostly of platformers that adhere to a specific (though somewhat evolving) core formula. There's not that much differences between Mario Bros and New Super Mario Bros, or Galaxy and 3D World, but FFVI and FFVII feel like different things altogether, and you could probably remove the Final Fantasy tag and no one would blame you for thinking they're two different games. Same with almost every other game on the franchise, aside from the originals (FF1, 2 and 3 being very alike, for example). One thing that definitively Mario trumps Final Fantasy is Sports games, so that's a huge base covered there.

The thing that made me put FF edging it was the fact that Mario lacks any meaningful MMORPG kind of game, on which FF has two and would certainly cover this (huge) demography as long as they could endure the setting and specific in-game mechanics. It's one of the biggest gaming markets, and it features an ever-evolving crowd of gamers. Now all they would need later is a MOBA, and that fits Final Fantasy much more than Mario, though it would not be impossible in either case considering it would be this specific franchise or nothing at all. xD