Here is the full list of all 20 Explodemon trophies - there are 20 bronze.
Destroy 15 enemies in less than five seconds
Beat the par times for all levels
Beat a par time for any level
Completely upgrade Explodemon and unlock all of his abilities
Close Explode the same green sticky 10 times
S-Rank all Squall levels
Complete Adventure Mode
Destroy 10 enemies in less than five seconds
Destroy 100 enemies by dunking them in acid
S-Rank all Arbor levels
Bring down a Vortex dropship
Perform a 3-Hit Combo
Destroy 1000 Vortex
S-Rank all Nibia levels
Explode 10 times in the air without touching the ground
Defeat Absorbemon without taking any damage
Complete any level with an A-Rank or higher
Destroy five enemies in less than five seconds
Level up to Max Level
Perform a Counter