How many continents are almost totally within the Western Hemisphere?

Only 2 continents, North America and South America, are mostly within the Western Hemisphere. Only the Aleutian Islands of Alaska extend into the eastern hemisphere.

By technical definition, parts of Western Europe, western Africa, and Antarctica are in the western hemisphere, as well as islands in New Zealand. The western hemisphere includes all areas west of the Prime Meridian at Greenwich, and east of 180° longitude (roughly the same as the International Date Line).