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