For example, can an electron and a photon become entangled?
An electron and a photon can become entangled -- http://www.nature.com/news/quantum-spookiness-passes-toughest-test-yet-1.18255 (demonstrated in that experiment).
I've read many times that everything in the universe is entangled with everything else, although I suspect you won't have CHSH inequality violation with most of that entanglement.
In fact an electron an photon being entangled is responsible for spontaneous emission.
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