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