Here is the full list of all 21 Incremental Adventures achievements.
Prestige on layer 5
Reach floor 100
Prestige on layer 2
Prestige for the first time
Reach floor 10
Reach floor 1000
Reach floor e10
Reach floor e100
Prestige on layer 10
Prestige on layer 20
Prestige on layer 50
Prestige on layer 100
Ascend for the first time
Ascend on layer 2
Ascend on layer 3
Ascend on layer 4
Ascend on layer 5
Complete an ascension trial
Complete five ascension trials
Complete ten ascension trials
Complete 25 ascension trials