Here is the full list of all 25 Recompile achievements.
Restore the TET Biome
Restore the HEX Biome
Complete TET Access Control without killing a single enemy
Collect all the memories in the TET Biome
Kill 32 enemies
Survive a lethal fall
Fully upgrade Traversal abilities
Fully upgrade Deletion abilities
Fully upgrade Hacking abilities
Fully upgrade all Abilities
Install yourself
Restore the ICO Biome
Restore the OKT Biome
Collect all the memories in the HEX Biome
Collect all the memories in the OKT Biome
Collect all the memories in the ICO Biome
Collect all the memories in the USER Biome
Complete the game with an ICO-Centric AI
Complete the game with an OKT-Centric AI
Complete the game with a HEX-Centric AI
Complete the game with a TET-Centric AI
Restore all Memories and complete the game with a Balanced AI
Defeat the Hyperspider
Complete TET Shield without killing a single enemy
Hack 16 enemies