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