ALGIERS — High above the sea in a heavily guarded villa, Algeria’s 82-year-old president, Abdelaziz Bouteflika, sits in a wheelchair — mute, paralyzed, barely able to move his hands. Hovered over by a flurry of attendants and family members, he has not uttered a single word in public, much less given a speech or interview, since a stroke in 2013.

Twenty miles away in the capital, Algiers, tens of thousands of demonstrators fill the streets every week loudly demanding his departure, and that of the extensive, ill-defined entourage around him that Algerians call simply the “power,” the nexus of high-ranking officials, wealthy businessmen and military officers who actually run the country.

The demonstrations, the largest in over 30 years, have grown larger every week and seem unstoppable. Algeria, the largest country in Africa and a rare pillar of stability in the Arab world, now faces an uncertain future.

The protesters’ demands are unambiguous: After two decades of undivided reign, Mr. Bouteflika, his clan, and his system must go.