I think that relativity and quantum mechanics would provide some good examples.
closed as not a real question by weiqure, pablasso, j.c., David Zaslavsky♦, Sklivvz♦ Nov 6 '10 at 7:54
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Laminar flow reversibility; gets straight from equations, but it is hard to believe it works in reality. Yet: http://www.youtube.com/watch?v=p08_KlTKP50
If you mean things I'm still scratching my head about, starting from the largest scale, it might be that we, life, the earth, the universe, (in Douglas Adam's words, "everything") is here at all. Different levels of the "Why something rather than nothing question" I suppose, but that's already too defined. We have only developed theories e.g. probability theory on branching trees, quantum chemistry, cosmology in the last 200 years to even partially explain the phenomenology. If someone told you the story from afar, your initial reaction would have to be "No way!" You could say it's not physics (no math and way too broad), but it's definitely physical phenomena.
For something smaller, quantum phenomena, e.g. quantization of spin axis. Is it as Zeilinger suggests ultimately because the electron spin carries only one bit of information?