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Alexander's horned sphere is a famous example, due to J. W. Alexander (1924), of a subset of space homeomorphic to a sphere whose exterior is not simply connected. In this construction the sphere is formed by recursively growing interlocking horns, each similar to the others. To get the complete sphere, the construction would have to be carried out to infinity. That is not possible, but this Demonstration can help get you started.

Note: you may rotate the 3D image as usual, but zooming can be done only by the slider. Also, panning does not work as expected, because of the way the program controls the image presented.

J. W. Alexander, "An Example of a Simply Connected Surface Bounding a Region Which Is Not Simply Connected," Proc. N. A. S., 10(1), 1924 pp. 8–10.