Here is the full list of all 27 Great Hero's Beard achievements.
Max out all princesses
Defeat the Chubby Tubby Warlord
10th Retirement
Defeat the Avalanche
Defeat Skelethrone King Slayer
Defeat the Sea King Neposeitune
Defeat Demon Lord Baal
5th Retirement
100'000 HP
200 ATK SPD
Rescue Fifth Princess
Defeat the Guardian Azriel
First Retirement
Reach Level 100
Rescue 4th Princess
Reach Level 80
Deal 1M Damage
100'000 Damage
Reach Level 60
Rescue Third Princess
100 ATK SPD
Rescue Second Princess
Reach Level 40
Deal 1'000 Damage
Rescue First Princess
Reach Level 20
First Fusion