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