Here is the full list of all 15 Froggy's Battle achievements.
Achieve very high DPS
Open a chest with 3 rare upgrades
Beat the final boss
Deal big damage
Dash 200 times
Die 10 times
Beat the twin boss
Jump 200 times
Die
Beat the first boss
Beat Hard Mode
Win in less than 5 minutes
Win Hard Mode in less than 7 minutes
Win with two shields
Win Hard Mode without losing a heart