The protein needs to pass through 3 double-layered membrane layers to reach its destination in the inner thylakoid space of the chloroplast.

The process starts with the protein being synthesized by ribosomes bound to the endoplasmic reticulum (ER). The protein passes through the ER membrane, which is the first double-layered membrane layer.

Next, the protein is transported from the ER to the Golgi apparatus, which is another double-layered membrane structure. This is the second double-layered membrane layer that the protein needs to pass through.

Finally, the protein is transported from the Golgi apparatus to the chloroplast, which contains a double-layered membrane system. This is the third and final double-layered membrane layer that the protein must cross to reach its destination in the inner thylakoid space.

Therefore, the correct answer is option 2 - the protein needs to pass through 3 double-layered membrane layers.

Alert - correct answer should be 4.

In addition to passing through the ER and Golgi apparatus, the protein also needs to pass through the outer membrane and inner membrane of the chloroplast, making a total of 4 double-layered membrane layers.