Let us define this as the largest observable density of a stable object, in order to exclude black holes which may have a very large (infinite) density at their centers or objects collapsing towards a black hole status.

If we restrict the definition in this way, then the answer should be the core of the most massive neutron star that we know about.

At present there are a couple of neutron stars with mass of about $2M_{\odot}$ (Demorest et al. 2010; Antoniadis et al. 2013. Depending on the exact composition and equation of state at their centres these should have densities of around $2 \times 10^{18}$ kg/m$^{3}$ at their centers and average densities of $\sim 10^{18}$ kg/m$^3$.

Note that these densities are around 3 times the density of a proton or neutron or 5-10 times the density of nuclei at zero pressure.

In principle, the density of a single electron is much higher.