Here is the full list of all 23 Cat Quest (JP) trophies - there are 7 bronze, 7 silver, 8 gold and 1 platinum.
MEOW!
Cleared all 52 dungeons
Obtained 66 armor or weapons
Completed the Main Quest
Completed all 62 side quests
Reached level 99
Cleared 25 dungeons
Defeated the Arcane Dragon
Reached level 50
Obtained 30 armor or weapons
Located all monuments
Obtained all 7 skills
Completed 30 side quests
Defeated the Ice Dragon
Obtained 4 skills
Located half the monuments
Defeated the Flame Dragon
Cleared 5 dungeons
Reached level 10
Completed 6 side quests
Obtained 2 skills
Obtained 3 armor or weapons
Located one monument