I pick these series because I can easily replay them over and over again.

Nintendo:

Super Mario

The Legend of Zelda

Sony:

Gravity Rush

Patapon (because Leadified told me)

Microsoft:

Banjo Kazooie

Conker (under the idea that these two will get good games of course)

Third Parties:

Atelier

Tales of

Other:

Metroid 

Pokemon

Nintendo:

- Pokemon (Sun and Moon, X and Y, red blue and yellow type): I don't think we have much games like these at all.
- Mario Bros platformer (if I can only choose 1 platformer franchise, it would be this one)

Sony:

- The Last of Us (It's got to have a great post-apocalyptic franchise, at least, and I choose TLoU)
- Bloodborne (I want a hardcore game that will make me die until I love it and only then git gud)
(I know Bloodborne is not a franchise yet, but shut up, let me dream).

Microsoft:

- Forza Horizon (Best arcade racer atm, IMO)
- Halo (there has to be a 1st person shooter, and it better be sci fi, and it better be good)

Third Party:

- Final Fantasy (I mean, pls, of course)
- Grand Theft Auto ( open world + madness + crime = a must)

Flex Party:

- Street FIghter (a fighting game, for those local multiplayer pwnings)
- Spideman (PS4: I think there has to be at least one superhero game, and I have faith in the upcoming Spiderman)

Man, this was tough. Great idea for a topic.

I cheat for a few of these.

Nintendo:
Super Mario
Zelda

MS:
Halo
Age of Empires/Mythology (defunct)

Sony:
Spyro (Not really Sony)
Infamous

3rd Party:
Final Fantasy (includes MMO?)
Dynasty Warriors (includes XL/ Empires?)
Elder Scrolls (MMO?)
League of Legends

Good mix of a lot of things, including 2 MMOs and a Moba to spend a lifetime in xD.

Zelda
Mario
Uncharted
Last of Us
Forza
Gears
Final Fantasy
GTA
Pokemon
CoD

Let's see...

-Zelda
-Xenoblade
-Ratchet & Clank
-Sly Cooper
-Banjo-Kazooie
-Forza
-Kingdom Hearts
-Dragon Quest
-Super Mario platformers
-Pokémon

1) Mario
2) Metroid
3) Uncharted
4) God of War
5) Halo
6) Forza
7) Elder Scrolls
8) Tactics Ogre/FF Tactics/Fire Emblem (cheating, I know)
9) Zelda
10) Mario Kart

Tough game... 20 franchises would have been easier. I'm missing some personal favorites from this list, either because they fit into a genre that was already over-represented or the recent titles had been lackluster.

Nintendo... 

• The Legend of Zelda: To satisfy action/adventure in its pure, unadulterated state, while actively having a recognizable icon to exploit at will.
•  Star Fox: This franchise cannot die. At least not yet! 

Sony... 

• Uncharted: This one needs no explanation. 
• Horizon: Seriously, just Horizon... 

Microsoft... 

• Forza: The best races and cars in the business. The FH series has become the new Need For Speed of the racing game world. 
• Halo: Shooters need a lot of lore and this one has an excess of useful and intriguing information. 

Franchises from third-parties... 

• Final Fantasy: Self explanatory. 
• Tales of: We need those games. We also better remind Bamco we don't like shovelware... 

• Grand Theft Auto: Your system will be doomed the moment Rockstar overlooks it... 
• The Elder Scrolls: Because they are awesome like that. 

1. God of War
2. Red Dead Redemption
3. Infamous
4. Batman Arkham
5. Forza Motorsport because I can't fit GT in the list
6. The Legend of Zelda
7. Devil May Cry
8. Pokemon
9. Doom
10. Forza Horizon

1. Mario Kart
2. Zelda
3. Ratchet & Clank
4. Wipeout
5. Forza
6. Fable
7. Dragon Age
8. GTA
9. Fallout
10. Mass Effect

Nintendo:
Fire Emblem, Metroid

Sony:
Crash bandicoot, Gravity Rush

Microsoft:
Halo, Forza Horizon

Third party:
Final Fantasy, Doom

Flex:
Minecraft, Super Smash