Here is the full list of all 21 Awesomenauts Assemble! trophies - there are 21 bronze.
Reach max level.
Win a game session with 2 friends splitscreen.
Play a game session with friends invited through the game session setup.
Taunt an enemy after he killed you.
Kill 1000 droids/creeps.
Destroy 25 turrets.
Kill an enemy Awesomenaut without buying any items.
Make the most kills of both teams in an online game session.
Win an online game session.
Taunt an enemy after killing him.
Win game sessions with 8 different characters.
Collect 35 Solar on a drop-pod drop.
Earn a total of 10000 Solar ingame (late game bonus does not count).
Kill 25 enemy Awesomenauts.
Unlock 3 characters.
Play against a team with the same Awesomenauts.
Get first blood.
Win a game in less than 10 minutes.
Get killed by a creep.
Have more than 500 Solar at once (late game bonus does not count).
Kill 3 enemy Awesomenauts without dying.