There are relatively few outright mysteries that remain - the Higgs Boson and the effects shaping the inflation of the universe (eg. dark mater) are classic examples of our time. I suspect that eventually, we will have a coherent explanation for all observable physical phenomena - it's not over yet by a long shot, but one day we'll figure it out.
What about this one: both general relativity and quantum mechanics are remarkably accurate in their respective areas (large masses and distances for GR, small ones for QM). One would expect some generalized theory that simplifies to GR for large masses/distances, to QM for small ones, and to Newtonian mechanics for intermediate ones. However, GR and QM are fundamentally incompatible. (Hint: time as inextricable part of spacetime in the one case vs time as a parameter, space as an operator in the other.)
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