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