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Hopf Fibration: Technically, the base space of the Hopf fibration is a 2-sphere (S^2), but since the 2-sphere can be thought of as a compactified plane or a disk with an added point at infinity, it could count.
Möbius Strip: The Möbius strip can be thought of as a fiber bundle over the circle S^1 with fibers that are intervals of the real line.
Twisted Cylinder: Similar to a Möbius strip but with the fibers being open intervals instead of closed loops.
The Klein Bottle: If you take S^1 as your fiber, the Klein bottle can be seen as a nontrivial fiber bundle over the circle.
Principal Bundles: The concept of a principal G-bundle, where G is a topological group, is a generalization of fiber bundles. For instance, the frame bundle of a manifold is a principal GL(n,R)-bundle, where GL(n,R) is the general linear group of invertible matrices, and n is the dimension of the manifold. As a more specific example, consider the tangent bundle of a disk, D^2. The frame bundle of D^2 is a principal GL(2,R)-bundle over the disk.
Do principle bundles not count as fiber bundles? A topological group is still a space, and GL(n, R) is even a Lie group.