Here is the full list of all 21 Incremental Adventures achievements.
Prestige on layer 2
Prestige for the first time
Reach floor 10
Reach floor 100
Reach floor 1000
Reach floor e10
Reach floor e100
Prestige on layer 5
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