Are there any gases more dense than liquids?
It depends on the conditions. Let's start decomposing your question in two related questions:

Denser gasses at SATP?

There are indeed some gasses that are quite dense. Sulfur hexafluoride has a density of 6.17 g/L  while tungsten hexafluoride of 12.4 g/L. But usually, they are not so dense compared to the density of liquids.

Lightest liquid at SATP?

The density of liquids largely depends on the atomic mass of the compounds. Hydrogen that is the compound with the lowest atomic mass has a density of 70.85 g/L, which is probably the lowest density you can find. 

So at room temperature is not possible but if you increase the pressure the density of the gasses will increase while the density of the liquid won't increase much because liquids are not appreciably compressible.  It is hence theoretically possible to achieve a gas with a greater density compared to that one of a liquid, and also the coexistence of the two phases should be possible in certain conditions but the pressure required won't allow you to observe it. 

Bubbles sinking down?

That won't happen also if there was a denser gas compared to a liquid. Bubble formation is quite a different process in this case inverting the gravity would be better! ;-)