With SeaWorld San Diego's aforementioned eight species, along with the African penguins at San Diego Zoo and the little blue penguins at Birch Aquarium, there's 10 penguin species that can be seen within a 15 mile radius. San Diego may just be the penguiniest city in the world!
To be specific, the eight species SeaWorld San Diego has are Emperor, King, Adelie, Chinstrap, Gentoo, Macaroni, Humboldt, and Magellanic. More information about what facilities have which penguin species can be found in this thread by @Chlidonias. Catalogue of Penguins in Captivity Worldwide
It’s definitely not breaking any records species-wise, but I think Cincinnati deserves a mention for keeping five species (king, southern rockhopper, Magellanic, African, little blue) across three habitats.