Physical DISPLAYS for Rec709 are defined as a gamma of 2.4. Note that this is different than the reciprocal gamma that the signal is encoded with. The sRGB spec may indicate ~2.2, but there's more to the story.
sRGB has a signal gamma (i.e. the stored image gamma) of ~1/2.2, though it is more precisely set as:
If this is fed into a monitor with a gamma of 2.4, for a total system gamma gain of 1.1. Why is this often the case? For sRGB per the IEC standard, the monitor's luminance is speced at 80 cd/m2. That was way back when, when CRT monitors had a hard time displaying brighter. Today LCD monitors can easily be set to 200 cd/m2 or more, and phones are available as high as 1200 d/m2 !
If you set your monitor brighter you are almost certainly going to set the monitor's gamma higher as well. Studies have shown that typical users have their monitors set over 160 cd/m2, and gamma much higher than 2.2 to 2.4 - one survey found that over 50% of users had the gamma at 2.5 and HIGHER!
The transfer curve of the sRGB signal is "close to" a 2.2 gamma, but it is actually a piecewise curve, that is actually a higher gamma at the top end of the video range and a lower gamma at the bottom of the video range.
Part of the issue is that ambient lighting has a very substantial effect on the perception of the colors and lightness of your monitor.
If you are in a bright environment with a dark monitor (peak white at 80 cd/m2 or lower), then you'll want your display gamma lower (i.e. 2.2) as that will brighten the image to compensate for the ambient, but unfortunately also reduce contrast.
If you increase the monitor's peak white level (say to 200cd/m2), you'll also want the gamma higher like 2.4 which will darken the midrange and increase contrast, which is needed due to the higher ambient lighting "washing out" the blacks.
Related: The eye's response to light is also non-linear — and that non-linearity is not the same throughout the visual range (i.e. even our non-linearity is non-linear!) and depends on total brightness.
For sRGB per the IEC standard, 80 cd/m2(includes veiling glare), gamma approx ~2.2, with an ambient of 64 lux (encode) or 200 lux (typical).
For sRGB per the SMPTE standard, 120 cd/m2.
For Rec709's BT1886 (monitor) and BT.2035 (viewing environment) the monitor gamma is 2.4, the whitepoint luminance is 100 cd/m2 and the ambient is 10 LUX (dark!)
And see this PDF for more.
So, how should you set your gamma? It depends on what you are doing, your environment, and if your monitor has an internal LUT (like a high end NEC or Eizo).
If you are using color management, then setting the monitor to a NATIVE gamma (i.e. relying on it's internal LUT), and profiling it with an XRite i1 Display Pro leads to good results.
That said, I usually have my NEC PA271W with a 14 bit internal LUT set to an L* curve.
And my Samsung 245t monitors needs hardware (front panel control) tweaking to get their "native" values in line with reality for a reasonably flat profile relative to sRGB.
In other words, your mileage may vary, along with your ambient conditions. But if you are using ICC profiles and color management, I find best results by making profiles, looking at the profile's curves, and adjusting the hardware controls on the monitor, profiling again, until the profile curves are relatively smooth.
The reason and theory is that you want the ICC profile to do as little transforming as possible.
(Edited for clarity, minor corrections, and to add a few other ideas.)