A PS3 can play at least every Final Fantasy from 4I to 13XIII-3 except for III and XII.
With backwards compatibility and PSClassics on PSN you can play:
Final Fantasy Origins - (Contains FFI and FFII)
Final Fantasy Chronicles - (Contains FFIV and Chrono Trigger)
Final Fantasy Anthology - (Contains FFV and FFVI)
Final Fantasy VII through IX
PS3 versions include X, X-2, XIII, XIII-2, and XIII Lighting Returns.
An original PS3 COULD have played XII. But the PS2 backwards compatibility which was discontinued early on.