Like others have said: Parasite Eve.
Square had an instant hit and a main character who immediately became recognizable in gaming.
Even though PE2 went with more action, it was still a quality sequel IMO. PlayStation 2 would have been a great system to put out a Parasite Eve 3 (likely in-between FFX and FFXII release) and keep the franchise rolling. I think sales would have been pretty damn good.
Nope.
Fans finally got a new game with 'The 3rd Birthday', but not only is it wasted potential by not being a big-deal console game (originally going to be a cell-phone game, for fuck sake!), but it also had Toriyama's shit-stained fingerprints all over it.
I would imagine Square-Enix really doesn't have a desire to continue the series thanks to The 3rd Birthday's taint.
I'm also going to say Silent Hill.
Toyama and KCET made a truly unique horror game with SH1. That continued via Imamura and his team with SH2, albeit thematically quite a bit different from the first one.
After SH2, it all went downhill (even though SH3 is still excellent technically) in the way of KCET Corporate wanting more control regarding the direction of the series to maximize sales.
Imamura had the idea that Silent Hill could be a 'Twilight Zone'-type series, i.e. every game being a completely different story/theme, but the only connecting threads would be the town itself and various aesthetic connections (mostly through Ito's art direction). He was even a bit ahead of his time by suggesting episodic releases at various points in the year telling a variety of creepy shorter stories. Unfortunately there just wasn't a means to distribute these kinds of smaller, discounted episodes at the time and these ideas were shot down, as one of the original stories was adapted to be an 'add on' to SH2 (Born from a Wish), one converted to be a direct sequel to SH1 (SH3), and another original IP was retooled to fit into the series (SH4).
After that, KCET simply didn't want to put the resources into the series, yet still wanted to profit from it. Their brilliantly talented in-house artists/designers/programmers were dumped into oblivion and Konami started outsourcing.
The series still has a small spark of life, but not much.
Regardless, the potential the series had back in the early 2000's was completely pissed away, IMO.