Here is the full list of all 36 Adventure Time: The Secret of the Nameless Kingdom trophies - there are 20 bronze, 11 silver, 4 gold and 1 platinum.
Use Flambo to defeat a Lub Glub
Win the race against Starchy
Find Slumber Princess
Stun 5 enemies with a single Bananarang
Find a renewable light source
Lie to Marceline
Appease Cinnamon Bun
Chop down 999 bushes
Shove a Fairy Convict into a Baggy
Squash a Pie Fairy
Find a Thump Shard
Find Lullaby Princess
Buy an item from Choose Goose
Complete the circle of love
Find a Gum Globe
Get briefed on the mission
Earn all trophies in the game.
Find Nightmare Princess
Locate the hidden Princess
Get Extra Ending 1
Get Extra Ending 2
Get the true ending
Max out your Thump Meter
Find every Baggy
Receive every type of Curse
Buy any variety of Trail Mix
Complete the Trading Game
Repair the wizard's door
Receive sword training
Make a mess of the Pottery Shop until you get something nice
Find every item in the entire game.
Defeat an enemy after hitting it with both Finn and Jake
Find every secret Gunter cave
Dance the night away in the Rave Cave
Attempt to speak to Stanley 5 times
Beat the game without upgrading Finn's Thumps, Sword, or Armor.