Here is the full list of all 55 Tokyo Xanadu eX achievements.
Clear the intermission -Coming of Summer-
Clear chapter 5 -Hidden in the Fog-
Defeat 1000 Greeds.
Clear chapter 4 -BLAZE: Firestorm of Memories-
Defeat 500 foes with Elemental kills.
Clear chapter 3 -Word of God-
Clear chapter 2 -Mysteries-
Defeat 100 Greeds.
Perform a 200+ hit combo.
Clear chapter 1 -The Eclipse-
Obtain an S rank in any stage.
Wander into the mysterious place of wonder.
Obtain all achievements.
Clear all stages with an S rank.
Max out Kou's "Wisdom, Courage, Virtue".
Obtain all information on friend page in NiAR.
Open all treasure chests.
Clear the game on Infinity difficulty.
Obtain all information on battle page in NiAR.
Obtain all information on recipe page in NiAR.
Obtain all information on book page in NiAR.
Obtain every single master core.
Complete all side-quests.
Amass a 500,000 yen fortune.
Pick the most fortuitous...fortune.
Defeat 2000 Greeds.
Use X-Drive 100 times.
Use X-Strike 100 times.
Use Strike Chain 50 times.
Defeat 500 foes with Critical hits.
Chain Kill 100 times.
Rack up 100 Aerial kills.
Inflict over 5000 points of damage.
Traverse 100,000 meters.
Get through a stage without taking any damage.
Qualify for the fast clear time bonus in any stage.
Forge an unbreakable bond with Asuka.
Forge an unbreakable bond with Sora.
Forge an unbreakable bond with Yuuki.
Forge an unbreakable bond with Shio.
Forge an unbreakable bond with Mitsuki.
Forge an unbreakable bond with Rion.
Forge an unbreakable bond with Shiori.
Forge an unbreakable bond with Ryouta.
Forge an unbreakable bond with Jun.
Forge an unbreakable bond with Towa.
Forge an unbreakable bond with Gorou.
Clear chapter 6 -Wing and a Prayer-
Clear chapter 7 -Hollow Quake-
Clear final chapter -Endless Night-
Clear epilogue -A Legend is Born-
Fully furnish Kou's room.
Clear the After Story on Calamity difficulty.
Use EX Skill 300 times.
Clear after story -From Here on Out-