Final answer:
The Golgi apparatus possesses multiple layers of membrane budded off of the nucleus and functions to store, package, and distribute proteins around the cell.
Step-by-step explanation:
The organelle that possesses multiple layers of membrane budded off of the nucleus and functions to store, package, and distribute proteins around the cell is the Golgi apparatus. The Golgi apparatus receives proteins from the endoplasmic reticulum and modifies them by adding sugars or other molecules. It then packages the proteins into vesicles and sends them to their final destinations within the cell or outside of the cell.