SO(3) is my favorite. It has so many weird properties. It pretends to be a 3-sphere, but it isn’t. It’s a subset of projective space. And yet it’s so useful in engineering. really the first physical non-Euclidean manifold we’re all introduced too, yet it isn’t an n-sphere, it’s non-commutative, non-constant curvature, yet it has so many nice properties like compactness, connectedness, it’s a Lie group, and clearly it has so many physical interpretations. Not to mention it’s completely represented by the space of orthogonal matrices with unit determinant.