Osmium or Iridium depending on who you believe.
Osmium wins because the specific gravity of Iridium is very slightly lower than Osmium.
Density, or volumic mass, is a measure of mass per unit of volume.
Specific gravity, now usually called "Relative density" is the number obtained in comparison to water. The higher it is the more dense it is.
1 cc (cubic centimetre) of water weighs 1 gram and therefore has a relative density of 1.
Osmium is 22.57 but Iridium is only 22.42
Iridium wins when the density is calculated using the "space lattice" method, which is considered to be more accurate, when comparing it to Osmium.
An atom of Iridium has 77 protons, 77 electrons and 115 neutrons. At.wt. 192.22
An atom of Osmium has 76 protons, 76 electons and 114 neutrons. At. wt. 190.2
Osmium has a crystal structure that is "Hexagonal"
Iridium is a "Cubic face centred" crystal.
This crystal structure is how the atoms align with each other to form the material itself.
A good example to understand this is with Carbon. A very dense and very hard form is DIAMOND where the carbon atoms are very tightly and closely bonded together. A very soft form is GRAPHITE, where the carbons atoms are very loosely packed and easily parted from each other.
The Iridium structure's "space lattice" is more compact than that of Osmium such that more protons, neutrons and electrons fit into a unit volume and thus it is denser.
You can see some crystal structures here: