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