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