The associated production of quarkonium with a vector boson or an additional quarkonium state are rare processes that provide new insight into QCD models of quarkonium production, but also provide new opportunities to study double parton scattering. We present new measurements of $J/\psi$ production in association with a vector boson or another $J/\psi$, in $pp$ collisions using the ATLAS detector at the LHC. Differential production cross-sections of these processes are measured, and a data-driven approach is used to estimate the single parton scattering and double parton scattering contributions to these rates.