It cant be any of the Gigaohm possibilities you offered as if it was, it would have to be mounted in free air between teflon standoffs as the bulk resistivity and surface resistance of FR4 PCB can be as low as 100 Meg m and 100 Meg/sq respectively. There's no point in soldering a resistor to two pads maybe 5mm apart if the resistor is a better insulator than the PCB between them!

One doesn't normally see resistors that are closer tolerance than 1% in simple transistor circuits. 1% resistors are absolutely ubiquitous and fill the market niche that in the old days was filled by the E12 10% resistor. Therefore, even if you are half blind and cant distinguish between white, grey and silver, and the camera colour rendering is also crappy enough to confuse us, 1 ohm 0.25% is very unlikely.
The choice between 680 ohms and 690 ohms is more difficult. Neither are E96 series (1%) preferred values. 690 is an E192 series (0.5%) value but one would expect a closer tolerance than 1%, and 680 is an preferred value in the E6, E12, and E24 series. There's a lot of design inertia in the trade, and experienced engineers still tend to use E12 series value whenever practical if the tolerance is non-critical so a number of resistor manufacturers offer E12 series values in their 1% resistor range so their products don't inadvertently get excluded from the BOM by less technically oriented purchasing departments who aren't authorised to adjust the value to the nearest one in a closer tolerance series.
Therefore I agree with basinstreetdesign, 680 ohm 1% is the most likely.
Its power rating is more difficult as better materials tecnology has enabled many resistor manufacturers to increase the max power rating available in traditional 1/4W and 1/2W body sizes. It *looks* like a 1/4W part but that's no guarentee nowadays. It certainly wont be 1/8W or 1/2W. If it *is* a 1/4W part, it should have less than 13V across it. As there's no sign of PCB discolouration, its unlikely to be running at max dissipation, so it probably has 10V or less across it. Is that consistent with its position in the circuit and the applied voltages?