ITU/EBU recommends that the end-to-end gamma of the video chain should be between 1.1 - 1.2 so we have a clearly defined target. It's commonly stated that video RGB is encoded with an exponent of 1/2.22 but the full function is:
v'=4.5*v for v < 0.018
v'=1.099*v^(.45)-0.099 for v >= 0.018
So what we want our displays to do is invert the function above and then apply a modest power of between 1.1 - 1.2.
What do we do in practice? We feed the display a linear input of 10% steps and (if we have a 10pt gamma control) try to get a nice flat fit to the equation v=v'^(gamma=2.2, 2.3, etc.) This completely ignores the fact that the real video signals we will eventually watch are encoded using the equation above.
So what are the consequences of the simplified power law calibration approach?
This is a plot of the result of feeding a linear signal which has been encoded using the BT.709/601 transfer function into a display with a simple power law response. Ideally we want a flat line somewhere within the cross-hatched region and you can see we don't get that. Most significantly the deviation below 20% stimulus can cause significant compression of the response as we go into black. You'll also notice that the response is not particularly flat until the upper stimulus levels. The implications of this is that contrast is being added at many of the levels which is not in the original video information.
I have been calibrating my display to a display gamma=2.3 and I find that I end up raising brightness a bit by eye to counter this effect when watching some material. If a 10pt gamma control is available a better approach is to actually calibrate to the BT.709/601 transfer function and this is an option in some of the calibration packages. Because it's not a simple power law, the way it's implemented (at least in HCFR) leads to a non-intuitive target gamma value (discussion here)
If you want to give this approach a try in HCFR use the "Camera gamma with standard offset" fitting function and targets of:
camera gamma = 2.44 for an end-to-end gamma = 1.1
camera gamma = 2.66 for an end-to-end gamma = 1.2
You'll find that to flatten out this gamma curve will require raising the 10%, 20%, and 30% controls (in decreasing amounts) relative to your standard calibration. I've seen people mention favoring a lower (power law) gamma at the low end and this achieves a similar result, but it's a much better match to how the video signal is actually encoded.
Before/After pictures, notice the overemphasis on shadow and loss of detail in the dress in the power law calibration.
power law gamma=2.3 End-to-End gamma is variable from 1.15 to 1.5
camera (inverse BT.709 transfer function) gamma=2.65 End-to-End gamma is flat at 1.2