My favorites Playstation franchises will have to be God of War, and the Jak & Daxter series. The reason I love the God of War franchise so much is because I first judged it by its cover. When I went to the store to buy new games, I saw God of War and thought that it was not interesting because of the way the cover art looked like (at the time I just arrived in the United States and did not know much about the Internet). But when my friend let me borrow it I fell in love with it!! The action was great, I fell in love with Kratos, and since then I have not missed on a God of War game.
My other favorite franchise among all of them have to be the Jak & Daxter series. I did not know about it until I decided to buy Jak 3 and see how it was. So I loved it too (mainly because my PS2 was programmed in Spanish so the characters were speaking Spanish, Daxter was the funniest one). Then I asked my friend if I could borrow the other parts. Did not get a chance to beat the first one, but Jak 2 was amazing and I loved it.
These are just 2 of my favorite franchises in Playstation (if I talk about them all it will come out the length of a research paper). I hope more amazing IPs come out for the Playstation, and if they are interesting enough, I will give them a shot.