Gigaminx and up: reduction
Reduce the puzzle to a megaminx, then solve it as a megaminx.
The gigaminx, teraminx, petaminx, examinx, zettaminx and yottaminx are megaminxes with more layers per face, as the 4x4 to 7x7 are to the 3x3. They are solved by reduction: build each face's centre, group each edge's pieces, then solve the result with megaminx steps and algorithms (wide turns move the inner layers with the face).
Notation follows big cubes: R turns the outer layer, 2R the second layer alone, Rw (or r) the outer layer and the next, 3Rw the outer three.
A gigaminx after a wide turn and a face turn.
Rw U
The yottaminx's top face and the pieces beside it.