A PS3 can play at least every Final Fantasy from I to XIII-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.