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