From the top of my head: (6 games or more)
NHL series: 10 titles (07-14)(16-17)
Call of Duty: 6 titles (COD 2, MW, MW2, Black ops, MW3, Black Ops 2)
GTA: 6 games (Gta 3, Vice city, San andreas, 4, 5, Chinatown Wars)
Pokemon: 8 games (Platinum, FireRed, SoulSilver, White, Black 2, Y, Alpha Sapphire, Moon)
Loads of franchises with 4-5 games played.

Define played?? Does spending 10 minutes with a game count??
Anyway the answer would be Pokemon or Mario, especially if we started to include the spinoffs.

I think you have to be a fan to have at least invested time into it.  For example, I've never invested any time into Pokemon - but I've at least tried a few.  That said, I wouldn't include them on my list.

I've counted them before and although I can't remember specific numbers it goes like this for the top 3

1. Sonic the Hedgehog
2. Mario
3. Final Fantasy

Mind this is for number of games completed, just games played would still look the same I think but anything in place 4 or below might change a lot based on just played vs completed.

1. Command & Conquer (11 games)
2. FIFA (8 games)
3. Age of Empires (7 games)
4. Assassin's Creed (6 games)
4. NBA 2K (6 games)
6. Total War (5 games)
6. Saints Row (5 games)
6. Championship Manager (5 games)
6. Need for Speed (5 games)
6. GTA (5 games)

Off the top of my head. Probably forgot something major.

My most played series should be Tekken. Played every entry since Tekken 3.

Right now its probably:

1. Harvest Moon/Story of Seasons
2. Hyperdimension Neptunia
3. Phoenix Wright: Ace Attorney
4. SMT/Persona
5. Animal Crossing

Pocket Monsters
Yu-Gi-Oh!
Kingdom Hearts
NBA Live
NBA 2K

Street Fighter (13+) - I, II, Champion Edition, Turbo, Super, Super Turbo, Alpha, Alpha 2, Alpha 3, III, III 2nd Impact, III 3rd Strike, IV and its "upgrades".

The King of Fighters (11) - 94, 95, 96, 97, 98, 99, 2000, 2001, 2002, 2003, XI.

I'm only going to count games that I have finished.....

1. Final Fantasy (9 games)
2. Tekken (8 games) - soon to be 9
3. Ridge Racer/Rage (6 gamess)
4. Mortal Kombat (5 games)
5. Metal Gear Solid (4 games)
5. Uncharted (4 games)