Here is the full list of all 61 Hero's Hour achievements.
Acquire a treasure elemental
Have a hero with an artifact in all equipment slots
Have three artifacts in a set
Fully develop a town
Use a Forlorn Cloister with your highest levelled hero
Defeat the black dragons in Skirmish mode
Defeat the ghost larvae in Skirmish mode
Defeat the red dragon in Skirmish mode
Win a game as Enclave (normal or harder)
Win a game as Tide (normal or harder)
Win a game as Wild (normal or harder)
Win a game as Horde (normal or harder)
Win a game as Lament (normal or harder)
Win a game as Pyre (normal or harder)
Win a game as Decay (normal or harder)
Win a game as Arcane (normal or harder)
Win a game as Pillar (normal or harder)
Win a game as Order (normal or harder)
Win a battle without taking casualties
Win a game without casting magic
Win a game without hiring a hero
Win a 5+ player game
Win a 7-player game
Win a game on the Patience map in less than 24 real-time hours
Win a game before week 3
Win a game in less than 15 minutes
Win a game in less than 30 minutes
Win a game without any artifacts
Win a Challenging difficulty game
Win a Hardcore difficulty game
Win a Normal difficulty game
Win an Easy difficulty game
Win a game as Earthen (normal or harder)
Win a game on Hardcore+ and with hamlet starting conditions
Defeat a near impossible enemy army
Defeat a challenging enemy in battle using only elementals
Complete a full artifact set
Recruit 666 Pyre units in one game
Acquire a dragon via Diplomacy
Have a hero reach level 20
Have a hero reach level 30
Achieve rank 5 in a hero skill
Have 100000 gold
Save 500 creatures using infirmaries and hospitals in a single game
Devour a Black Dragon
Learn an adventure map spell at an Archmage Tribunal
Create an Idol
Convert a town to a faction you did not start as
Put 200 power worth of units to work at a Slave Pens
Bloodwarp a Hatchling into a Seraph
Enter combat against an enemy player with at least 2000 power worth of army
Control 10 overgrown mines
Enter combat with a hero that uses the most overpowered Rune
Have a hero with a sum of attack, defence, knowledge and spellpower above 100
Cast Shadowcasting while your hero has 50 or more spellpower
Find an obelisk treasure with just 1 clue
Encounter 95% of all rare creatures
Encounter 95% of all artifacts
Encounter 95% of all spells
Encounter 95% of all adventure buildings
Achieve 95% of all achievements