Here is the full list of all 37 The Slimeking's Tower achievements.
Midas Unlocked
Reach the basement
Jade Unlocked
Reach Theta
Reach Tau
Reach Ohm
Reach Pi
Reach Omega
Attack a mimic before it's transformed
Igneous Unlocked
Reach Gamma
Defeat Lodeslime
Reach Epsilon
Maiyan Unlocked
Reach Delta
Reach Beta
Get your first boss key
Reach Sigma
Play Adventure Mode for the first time
Unlock Hairry's secret item
Beat the portal
Reach The Portal
Dots Unlocked
Mind Unlocked
Void Unlocked
The Collector Unlocked
Prayer Unlocked
Dots Completed
Mind Completed
Maiyan Completed
Midas Completed
Igneous Completed
Void Completed
Jade Completed
The Collector Completed
Prayer Completed
Hairry Completed