Here is the full list of all 63 Spell Disk achievements.
Climb over 30 floors in infinity dungeon
Stay alive for 5 minutes in Arena
Buy all three items in the shop
Open the gate to the Summoning Chambers
Defeat all Barrack bosses
Defeat all Crypt bosses
Defeat all Grotto bosses
Win with a Magic Book
Survive as a Cursed
Survive as a Gazer
Survive as a Trembler
Survive as an Incantor
Survive with a Hammer
Survive as a Chronomancer
Survive as a Loafer
Survive with a Spear
Survive with a Sword
Survive as a Cryomancer
Survive as a Giant
Survive as a Sprinter
Survive with an Axe
Survive with a Scythe
Survive as a Pyromancer
Survive as a Dwarf
Survive as an Electromancer
Survive as a Visionary
Survive as a Refresher
Survive as a Alchemist
Survive as a Determined
Survive with a Katar
Survive as a Deranged
Survive as a Rogue
Survive as a Duplicator
Survive as a Collector
Survive as a Wayfarer
Survive as a Novice
Survive with a Staff
Win with Danger Level 9
Win with Danger Level 8
Win with Danger Level 7
Win with a Spear
Win with Danger Level 6
Win with a Scythe
Win with Danger Level 5
Win with a Hammer
Win with Danger Level 4
Win with an Axe
Win with Danger Level 3
Win with Danger Level 2
Win with a Katar
Win with Danger Level 1
Win with a Sword
Win with a Staff
Win with Danger Level 0
Win with 28 items equipped
Win without expanding a item slot
Listen to the administrator in Imperial Palace
Defeat the Dungeon Architect three times
Win with a Scepter
Survive with a Scepter
Survive with a Magic Book