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