Official Site: Global Index
Square Enix (formerly Enix) are the publishers for several of the tri-Ace games, up to 2009 they were the only publisher tri-Ace used. However since then tri-Ace have also published games with Sega & Konami.
Enix was originally formed back in 1985 under the name of Enix Corporation, on the 1st of April 2003 Enix merged with Square Co., Ltd to become what is the now well known Square Enix.
The biggest non tri-Ace series which Enix published would be the Dragon Warrior (now Dragon Quest) series. It's no coincidence that the last two titles published under Enix was Dragon Quest Monsters: Caravan Heart & Star Ocean: Till the End of Time (the Japanese original version).
The biggest name title for Square Co., Ltd prior the merger was without a doubt the Final Fantasy series. In Japan the two most popular RPG series are Final Fantasy & Dragon Quest, arguably making them the the most successful RPG producer worldwide.
Here is a breakdown of which company has published the tri-Ace games to date.
As Enix, prior the Square Enix merger: