Quantum mechanics, as in understanding the mathematics behind its foundational issues, and not just as in computing the spectrum of the Hydrogen atom (though that's good too).
It's hard to think of a topic that shakes one's image of the physical world harder than quantum mechanics. General relativity is easy to digest once you are not scared of things like manifolds. Quantum mechanics remains a challenge to one's worldview no matter how hard one tries to get used to it. You cannot count yourself scientifically literate if you were not exposed to the foundational issues of quantum mechanics.
And it's a math course at least as much as a physics course. The pre-requisites are basic probability and logic and complex numbers and basic Hilbert space theory, and the content is philosophy and non-commutative probability theory and (may as well, at the end) some spectra of some differential operators.
Mermin article "Is the moon there when nobody looks? Reality and the quantum theory" was an eye opener for me, the year after I finished my undergraduate studies.
