According to the answer to the Chemistry SE question "Can alloys or ceramics have higher density than elemental solids?", it is possible that heavy alloys could be marginally denser than their parent materials, but no compelling example is given.
It is true that compounds can have higher density than their constituent elements. For example, sodium chloride (NaCl) is more dense (2.17 g/cm$^3$) than metallic sodium (0.968 g/cm$^3$) or liquid chlorine (1.56 g/cm$^3$), but this is because very tight NaCl ionic bonding reduces the inter-atomic spacing in the compound compared to the pure elements. All the elements that might make a heaviest alloy, (e.g. osmium, iridium, platinum, …), however, are chemically similar and should not bind to each other significantly more tightly than to themselves, so their alloys are unlikely to be measurably denser than the pure elements.
One might also try to increase density slightly by creating an interstitial alloy where smaller atoms (e.g. H) are inserted into the spaces of the crystal lattice of larger atoms (e.g. Pd), but the interstitial atoms typically expand the lattice such that the density changes little or actually decreases.
There was some hope that seriously denser (relatively) stable elements might exist in the Island of Stability, but searches so far have been unsuccessful.
