1. Super Mario

2. The Legend of Zelda

3. Tales of

4. Kirby

5. Metroid

6. Pokemon

7. Fire Emblem

8. Atelier

9. Pikmin

10. Castlevania

Here's 10 in no order:

Fatal Frame

Super Mario Bros

Super Smash Bros

The Legend of Zelda

Ace Attorney

Metal Slug

Trauma Center

Mega Man

Fire Emblem

Pokemon

Here's my list could be longer but stopped at 20, lol.
1. Soul's Games
2. Legend of Zelda
3. Metal Gear
4. Metroid
5. Final Fantasy
6. The Elder Scrolls
7. Fallout
8. Resident Evil
9. Sly Cooper
10. Mega Man / X

Honorable Mentions
11. Mass Effect / Dragon Age
12. Mario games
13. Xenoblade
14. Dino Crisis
15. Castlevania
16. Duke Nukem
17. Tomb Raider
18. Prince Of Persia
19. Left 4 Dead
20. One Piece Games

I started making a list, but I realized that my favorite franchises are the ones that dominate whatever genre they are in. Think of an genre, and the current game that sits on top of that genre.

Open world sandbox: GTA
Narrative driven: The Last of Us
FPS: Was CoD for a decade, currently this one is moving to Player Unknown Battlegrounds
RPG: The Witcher 3
Action Adventure: Zelda

I think the important genres change, and obviously the game in that genre that is the most important also changes, RPG was certainly Final Fantasy for maybe even the majority of the genre's existence, but western RPGs have become so important and Witcher is the prime target of anyone making a western rpg today, that it is certainly leading game design in that genre.

GTA is something everyone is always trying to emulate, it's one of the most prolific series to have ever existed, and helped propel the PS2 to it's insane install base. GTA3 had such a huge impact that the entire western industry has been chasing the overly expensive AAA market and completely ignored AA game development.

Zelda has lead the industry with game design for years, OoT is a road map for the last 20 years of game design in a 3D story. BotW has finally revitalized it's place back ontop of this genre.

The Last of Us is largely considered the peak of last gen, and is a masterpiece in story telling. Notable series also include Mass Effect and Halo.

CoD ran the FPS world for a decade, but before that, it was counter strike, and before that was quake and doom, this genre changes a lot and there are many notable games, but I feel like Player Unknown Battlegrounds is taking the genre in a new direction that we haven't seen since Counter Strike. Of course I'm ignoring Overwatch because I feel like it is more than just an FPS, it feels almost like a moba, and thus I didn't include it, but if you do, it's obviously on top and maybe should be, however PUBG is not something to overlook, it will change this genre forever.

Hmm good question...

Right now I'd probably go like this:

10) Gears of War
An amazing trilogy of third-person shooters led into a surprisingly awesome fourth mainline installment on XOne. Great shooting, spectacular action, fun co-op, addictive horde mode, and lots of offline options make this series a winner.

9) Mario Kart
This evergreen kart racer continues to impress, especially after its eighth and best installment.

8) Castlevania
Whether on NES, TGX16, PS1, GBA, or DS, Castlevania delivers the goods. It works as both a platform-action game and as a back-tracking action-RPG.

7) Resident Evil
The series has a few misses but overall it stands as the industry's hallmark survival horror franchise. The most recent installment was a triumphant return to form

6) Donkey Kong
Whether it's Donkey Kong Jr on NES or Donkey Kong Country Returns on Wii, the DK series has delivered stellar, challenging platforming action.

5) Halo
The series is in a steep decline but that doesn't stop me from adoring the years between 2001 and 2010. This is the best shooter franchise yet made, one that prized level design, AI, and improvisational gunplay.

4) Final Fantasy
This will probably rise in the rankings as I play more titles. So many thoughtful, legendary RPGs under this label, including the magnum opus FFVI.

3) Metroid
Although it's gone dormant a couple times, it remains one of my favorites. Super Metroid is the best of its kind, and the Prime trilogy is a master class in world design and environmental storytelling.

2) Super Mario
This series is constantly battling it out with #1. These are the two best series from the best game maker in the world. Super Mario is great both as a drum-tight 2D platformer or as a more open-ended 3D action game. Level design, music, and sheer creativity carry this series to the top.

1) The Legend of Zelda
Masterpiece after masterpiece for the last three decades, including maybe the best game ever made earlier this year. The Zelda formula is such an awesome baseline, and then new items, mechanics, and hooks are added for each new installment.

You included some in your list I didn't have room for, like DK and Kirby D: Also really good series.

1. Xeno (gears, saga, and blade)
2. Zelda
3. Mario
4. Final Fantasy (only 1, 3-8, Tactics, X, and the gameboy games that are technically other franchises)
5. Harvest Moon (now Story of Seasons and Rune Factory, screw Natsume's current "Harvest Moon" games)
6. Paradox grand strategy games
7. Fire Emblem
8. Kirby
9. Ace Attorney
10. Animal Crossing

Those happened almost on accident actually. My whole love of horror came from a friend I met incollege using me as a guinea pig to try all kinds of horror games, which slowly made me grow an appreciation of the genre. That same guy got me into SNK's line of games, where I also discovered and became a huge fan of Metal Slug. And Trauma Center I only found out because my friend happened to rent it. I tried it out, thinking it would be a stupid game, but I actually found it really fun, and bought the later installments.

1. Dark Souls/Bloodborne

2. God of War

3. Metal Gear

4. Uncharted

5. Gran Turismo

6. Battlefield

7. Gears of War

8. Ace Combat

9. GTA

10. ME

{1} Resident Evil
{2} Paper Mario
{3} Pikmin
{4} Super Smash Bros
{5} Sonic The Hedgehog
{6} Animal Crossing
{7} Professor Layton
{8} LittleBigPlanet
{9} Mario Party
{10} 3D Super Mario

[Note1 This list was difficult to make due to the way my mind thinks. I can distinguish enjoyment on a scale of "I hate it", "Meh", "I like it", "It's great", and "it's awesome" but beyond that, it sort of runs together. (As in, all things deemed as "meh" are the same, I don't enjoy one more than another)

[Note 2] Resident Evil is only that high because of the first 5 games. Paper Mario is pretty much tied with it. Pikmin is just as good but only has 3 games for now so it's lower.

[Note 3] Sonic is only there for the 2D & adventure games

[Note 4] The following series could make the cut in the future but I haven't played enough games to say:

*The Last Of Us
*The Evil Within
*Donkey Kong Country
*Splatoon
*Crash Bandicoot
*inFamous
*Heavy Rain, Beyond, Indigo (would be on list if we count them as one series)