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