Fans of classic Squaresoft RPGs have had a smorgasbord on the Google Play Store as of late, but it's all been remakes and re-releases. The first "new" Final Fantasy game to come to the platform (unless you count some of the simple stuff like Final Fantasy All The Bravest, which you shouldn't) is Final Fantast IV: The After Years. It's a sequel to the old FFIV (from 1991) released for Japanese mobile market in 2008 before making it to the Wii in 2011.
Since it was originally released as a series of downloads, The After Years is split up into ten playable chapters, four in the main story and six side stories. The game has been given a full 3D makeover not unlike the Nintendo DS re-releases, and of course it's got full touchscreen support. Final Fantasy veterans will want to know that the The After Years utilizes the active time battle system, with additional quirks provided by lunar phases and full-party attacks.
Square Enix has a high opinion of its titles, and The After Years is no exception: the price comes in at a whopping $15.99. That said, this might actually be worth it for hardcore Square fans, especially if they enjoyed the original FFIV (which is also available on Android). The new title uses Google Game support for achievements. Unfortunately it doesn't work with ART in Android 4.4; if you're on KitKat, enable the default runtime before starting.