Good news! Square Enix has announced that it is no longer limiting the MMORPG's free trial to 14 days.
According to the FF14's most recent patch notes, "The 14-day limit of the free trial has been removed." Below are the full trial notes:
The Japanese patch notes are a bit more specific, explicitly stating that the 14 day period has been changed to "indefinite" (14日間から無期限に変更されます).
The free trial still has certain caps and restrictions, but this certainly seems like a welcome way to incite players to give the game a look.