The final Plasmid slot is unlocked towards the end of the game as part of the story (along with the final of the 11 plasmids)
Puchase Plasmid slots at the Gatherers Garden whenever you see them available. Theyre actually quite cheap. I would advise adopting and rescuing all of the Little Sisters - as well as equipping the two tonics which increase and speed up the Adam gathering process.
I had one slot left to fill from Dionysus park so you can get most of them by mid game. As I say though that final slot is unlocked as part of the story so dont worry.