Here is the full list of all 53 The Legend of Heroes: Sen no Kiseki III achievements.
Earned all achievements.
Achieved instructor rank S.
Achieved an academy grade of Prestigious.
Completed the game on nightmare difficulty.
Completed all quests, both at the branch campus and during field exercises.
Opened every treasure chest.
Completely filled the battle section in the notebook.
Completely filled the character section in the notebook.
Completely filled the books section in the notebook.
Completely filled the cards section in the notebook.
Completely filled the recipe section in the notebook.
Completely filled the fish section in the notebook. (All 25 types of fish + Rod Lv. Max)
Collected all radio material and reported it to Munk.
Took all landscape photos and reported them to Vivi.
Gathered all 28 varieties of master quartz.
Acquired over a million mira.
Raised a master quartz to level 7.
Won 100 battles.
Won 600 battles.
Initiated 300 advantage encounters (including double and triple advantage).
Used 100 S-Crafts in battle.
Won a battle with a tactical bonus of x3.0 or greater.
Inflicted Break over 800 times.
Raised any two characters to Link Level 5.
Used over 500 follow-up attacks in battle.
Used Rush in battle 100 times.
Used Burst in battle 50 times.
Used Brave Order in battle over 100 times.
Earned an S-Rank on any chapter.
Forged a strong bond with Juna.
Forged a strong bond with Kurt.
Forged a strong bond with Altina.
Forged a strong bond with Musse.
Forged a strong bond with Ash.
Forged a strong bond with Towa.
Forged a strong bond with Alisa.
Forged a strong bond with Elliot.
Forged a strong bond with Laura.
Forged a strong bond with Machias.
Forged a strong bond with Emma.
Forged a strong bond with Jusis.
Forged a strong bond with Fie.
Forged a strong bond with Gaius.
Forged a strong bond with Millium.
Forged a strong bond with Sara.
Forged a strong bond with Elise & Alfin.
Completed the orienteering exercise.
Completed the prologue.
Completed Chapter 1.
Completed Chapter 2.
Completed Chapter 3.
Completed Chapter 4.
Completed the final chapter.