I’m glad that you said we can pick an entire series, because this makes the choice so much easier.
I think that my first choice would be the Metal Gear Solid series. Not only are the Metal Gear games my favorite of all-time (and one of the longest running video game series to boot), but their cutscenes are so long it’s practically like taking a set of movies with you as well! Bonus points.
I suppose the number two spot goes to the Super Mario Bros series. I don’t think there is such a thing as a bad Super Mario game, and with so many games to choose from, they’re bound to provide weeks of entertainment.
And the last spot would go to the Pokémon series. Not only are there dozens of Pokémon games, but truth be told, I’ve actually never finished a single one. I always beat all the gym leaders but have never gone on to face the Elite Four or complete the Pokedex. Being stranded on a deserted island would be the perfect opportunity to play through all the Pokémon games, to completion.