Here is the full list of all 34 Chocobo's Mystery Dungeon EVERY BUDDY! trophies - there are 16 bronze, 12 silver, 5 gold and 1 platinum.
Completed a dungeon with every type of monster as a buddy.
Defeated Shinryu.
Fused an ALL brand.
Completed Chapter V.
Maxed out one job.
Defeated 6 monsters with a single attack by Chocobo.
Raised Chocobo to Level 50.
Befriended a boss monster.
Completed Chapter IV.
Befriended 60 monsters or characters.
Completed Chapter III.
Befriended 20 monsters or characters.
Completed Chapter II.
Completed Chapter I.
Completed the tutorial.
Collected all trophies.
Collected one of every type of collar.
Collected one pair of every type of talons.
Collected one of every type of saddle.
Collected one of every type of item.
Befriended all monsters and characters.
Completed all Dungeons of Remembrance.
Completed all dungeons.
Reached Floor 100 of the Insatiable Hunger dungeon.
Reached Floor 500 of the Insatiable Hunger dungeon.
Increased your gear's level to +99.
Used every type of magicite.
Completed a 20-floor dungeon without triggering a trap.
Defeated 10 monsters with a single attack by Chocobo.
Defeated Omega.
Maxed out Chocobo's level.
Acquired all jobs.
Completed a dungeon with every job.
Maxed out all jobs.