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Which is the densest gas known to exist under normal "shirt sleeve" conditions: room temperature, one atmosphere, won't explode on contact with air, won't kill everyone in the building in trace amounts, etc.

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Perfluorobutane is inert and has almost twice the density of sulfur hexafluoride. It is non-toxic enough that it is used in fire extinguishers and injected as a contrast agent for ultrasound. Boiling point: $-1.7\ \mathrm{^\circ C}$.

Perfluoropentane is similar and rarer but somewhat higher density $(\sim13\ \mathrm{kg/m^3})$ in proportion to its higher molecular mass. Its boiling point is $28\ \mathrm{^\circ C}$ (uncomfortably warm, but your thermostat can go that high). This is the densest gas that strictly meets all the criteria in the OP.

If we relax the criteria a bit:

Perfluorohexane is just over the boiling point limit at $56\ \mathrm{^\circ C}$, but it has a molar mass of $338\ \mathrm{g/mol}$ which makes it slightly denser in gas form than tungsten hexafluoride ($\ce{WF6}$). It's also inert and non-toxic, unlike $\ce{WF6}$.

The ultimate gas density would be uranium hexachloride using depleted uranium and $\ce{Cl-37}$, with a molar mass of $460\ \mathrm{g/mol}$, which makes it over 50 % denser in gas form than tungsten hexafluoride, and 3 times denser than sulfur hexafluoride, but it has a boiling point of $75\ \mathrm{^\circ C}$, decomposes on contact with air, is toxic and slightly radioactive.

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    $\begingroup$ I found a bunch of sites referencing OsF8 properties, but no confirmation that it had really been synthesized yet. $\endgroup$ – Jonathan Ray Feb 8 '17 at 10:07
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Actually the densest gas which has an actual use is Tungsten hexafluoride, clocking in at an astounding 13 grams per liter. Tungsten hexafluoride isn't as well known as its lighter cousin, sulfur hexafluoride, because it has a narrow range of uses. It is useful in the field of electronics manufacturing as it can be used to coat circuit boards with Tungsten. Also, as fun it may sound, don't try to dip your hand in it or try to swallow it, as it's quite corrosive. Besides being corrosive itself, it also produces hydrogen fluoride, a nasty chemical, when it comes in contact with any moisture. Although it's a pretty unsafe chemical, it's cousin, sulfur hexafluoride, is safe enough to inhale in reasonable amounts.

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I would guess radon is the densest gas ($9.73\; \text{kg/m}^3$) that is not directly lethal. It is radioactive however, emitting alpha radiation, so you don't want to breath it in. Given that it is much heavier than air, as long as you don't hug the ground it think it would be possible to be in the same room without killing you.

From the fully non-lethal stuff, the first that comes to mind is sulfur hexafluoride, $\ce{SF6}$ ($6.17~\mathrm{kg/m^3}$). It is the stuff that people use to artificially make their voice lower (similar to helium making it higher) and it has a 0000 NFPA rating.

In principle you can dream up your own densest gas, because the density of a gas at given temperature and pressure is directly related to its molecular mass. So the higher the molecular mass of the gas, the denser it is.

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    $\begingroup$ But in principle you can't just dream up your own densest gas by adding molecules to the compound (to increase its molecular mass or density), because doing so will most likely give you a liquid or solid at standard conditions. Whether a compound or element is a gas at certain conditions depends not just on its density (or molecular mass), but many other chemical and quantum properties. $\endgroup$ – DrZ214 Jul 1 '15 at 5:41

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