What would happen if we were to compress the sun to the size of Lexington and then add a bit more mass? Would this object expand, contract, or stay the same size? Nuclear astrophysics research aims to answer this and other seemingly simple questions. This talk will focus on the nuclear physics of the outer layers of accreting neutron stars. I will show how properties of atomic nuclei impact dense matter constraints obtained for these objects. I will also discuss recent experimental work dedicated toward constraining properties of key nuclei.