HDMI 1.2 actually works up to 1.65Gbps. The 165MHz is the pixel frequency. There are 10bits/pixel (using the standard 24bit color - 8 bits per RGB lane, then TMDS coding that adds 2 more bits per lane). Note that the minimum 20% to 80% transition time for HDMI 1.2 is 75 picoseconds. So even HDMI 1.2 can contain some very high frequency energy.

So the characteristic impedance (Z0) of a transmission line is the square root of (L/C). Using a large gauge only increases the capacitance minutely. Capacitance is related to the closeness of the two wires, the dielectric constant of the material between the wires, and the amount of surface area where the electric field developes. Increasing the gauge only increases the surface area component, and increases capacitance. But note that increasing the gauge decreases inductance. Inductance relates to the area between the wires, the dielectric, and the current density flowing down the wires. When gauge is increased, the inductance would decrease as long as the spacing between the wires was kept the same. So to keep the characteristic impedance the same (100ohms differential for HDMI TMDS pairs) the spacing would need to increase (to reduce capacitance). since inductance would decrease with the larger gauge.

So for the 10 gauge example. You could run high frequency signals down a 10 gauge cable if the geometry creates a matched impedance. This of course would require a large amount of dielectric material. Speaking of this situation. Imagine the transmission lines that run to television antennas. These are huge gauge "cables" that carry thousands of watts to the antennas. Check out:
http://www.dielectric.com/broadcast/catalogs/FLEXLine.pdf
These are very large - not sure the fit on the gauge scale - cables that operate with low loss up to nearly the GHz range.