**Core Concept**
Amoebiasis is a parasitic infection caused by Entamoeba histolytica, which can affect various organs beyond the gastrointestinal tract. The extra-intestinal manifestations of amoebiasis are crucial to recognize, as they can be life-threatening if not treated promptly.

**Why the Correct Answer is Right**
The most common extra-intestinal manifestation of amoebiasis is Amoebic Liver Abscess (ALA). This occurs when the amoeba enters the bloodstream and lodges in the liver, causing an abscess. The pathogenesis involves the amoeba invading the liver through the hepatic portal vein, leading to necrosis and abscess formation. The liver is a common site for extra-intestinal amoebiasis due to its rich blood supply and the presence of amoeba-tropic macrophages.

**Why Each Wrong Option is Incorrect**
* **Option A:** Cutaneous Amoebiasis is a rare manifestation, typically occurring in individuals with compromised immunity or in areas where the amoeba is endemic. It is not the most common extra-intestinal manifestation.
* **Option B:** Amoebic Meningoencephalitis is a rare and severe complication of amoebiasis, usually seen in immunocompromised patients. It is not the most common extra-intestinal manifestation.
* **Option C:** Amoebic Pneumonia can occur in rare cases, but it is not the most common extra-intestinal manifestation of amoebiasis.

**Clinical Pearl / High-Yield Fact**
Amoebic Liver Abscess can mimic other liver conditions, such as pyogenic liver abscess or liver cancer, making it essential to consider amoebiasis in the differential diagnosis of a liver abscess, especially in travelers to endemic areas or in individuals with a history of gastrointestinal amoebiasis.

**Correct Answer: C. Amoebic Liver Abscess**