I consider the following pretty much essential PS1 titles, add any that you missed.
- FF series, any will do.
- Ace Combat series, again any will do.
- Ape Escape (totally made of win)
- Silent Bomber. Despite the fact that every copy I've ever owned has been broken and thus I've never seen the ending.
- Crash Bandicoot series
- Metal Gear Solid. Before it got so damn confusing you need an FAQ to keep track of the story.
- Resident Evil series
- Worms
- Hogs of War
- Time Crisis
- DDR
- Wild Arms
- Legend of Dragoon
- Galaga Destination Earth.
- Alundra