Here is the full list of all 15 No Heroes Allowed: No Puzzles Either! trophies - there are 11 bronze and 4 silver.
Create your first monster.
Complete your first combo.
Complete the Monsterpedia.
Complete the Heropedia.
Complete Secret Stage 8-8.
Complete Stage 8-8.
Complete Stage 7-8.
Complete Stage 6-8.
Complete Stage 5-8.
Complete Stage 4-8.
Complete Stage 3-8.
Complete Stage 2-8.
Complete Stage 1-8.
Input the secret command.
Complete a stage collaboratively.