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