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