This image (not shot by me) is shot using 70-300mm f/4.5-5.6 lens, but the EXIF shows the focal length to be 302 mm. How is it possible to be shot at a focal length (302 mm) greater than the permissible maximum (300 mm)?
When the engineers are designing a lens, the 70-300 is a target focal range they design to, but it's not important that they hit it exactly as long as it covers the advertised range. As they tweak the lens characteristics to get a suitably sharp and quality image for the target cost, the actual focal lengths may change slightly. Eventually, when everything is "good enough" it's time to mass produce it, as it's not worth the time to have the staff readjust the design to stop right at 300.
It's not at all uncommon to have lenses, even prime lenses, have an actual measured focal length to be slightly different than the listed focal length. Nobody is really going to care if their 50mm prime lens is actually 48.9mm or 52.1mm, but it's close enough to compare it to other "50mm" primes, and searching for a lens in a catalog where it listed 48.9 instead of nice round numbers like 50 would be even more obnoxious than lens shopping already is. When you are looking for a 35mm lens, you are really looking at a class of lenses where the focal length is 'close enough' to 35mm.
Then, adding to all this, the focal length of a lens can often change with the focus distance, and for photography lenses this is almost always the case. This is known as "focus breathing", and only for cinema lenses does anyone bother to design the lens to maintain the same focal length throughout the focus distance range. Supposedly the focal length is supposed to be specified with the lens focused at infinity, but I don't know if that's true.
whatsisname mentions one possibility, which is that the lens in question actually allows focal lengths slightly beyond its nominal range.
Another possibility is that the photographer was using a teleconverter, which multiplies the focal length of the lens by some factor. Nikon offer 1.4x, 1.7x and 2x teleconvertors which, respectively, would turn the 70-300mm lens into 98-420mm, 119-510mm and 140-600mm. When I use my 1.4x teleconverter on my Canon camera, the EXIF data doesn't record that fact so, for example, if I use it with my 70-200mm lens, the EXIF reports that I was using the 70-200mm with a focal length somewhere in the range 98-280mm. I don't know if Nikon teleconverters behave the same way, or if they get reported in the EXIF.
If this is the explanation, it's just a coincidence that the reported focal length was slightly higher than 300mm. Given that the photographer was still (almost) within the focal length range of the lens without a teleconverter, they'd have got a slightly higher-quality image by leaving out the teleconverter. The extra optics results in a slight degradation of image quality.
Actually, I just checked the EXIF of the image more carefully and it contains "Min Focal Length - 71.3 mm, Max Focal Length - 302.0 mm". So the actual answer is that the lens is only approximately 70-300mm. However, I'll leave the teleconverter explanation since, although it's incorrect in this case, it does apply to the equivalent question about other images.
The focal length is the distance measured from the rear nodal of the lens array to the film/sensor when the camera is imaging a distant object. This will be an object at infinity ∞ (as far as the eye can see). At any other distance, the lens to film/sensor distance will be elongated. At unity (life-size 1:1) the distance lens to film/sensor is 2X the focal length. Thus when imaging objects nearer than infinity, this longer projection distance is termed “image distance” or "back focus" rather than focal length.
Additionally, lens makers are not generally required to hit the focal length “dead-on”. If fact each lens, coming off an assembly line will be unique and variance can be as much as 3%. Additionally, the data provided by the camera software might be the back focus distance rather than the focal length.
Suppose the lens is dead on and is exactly 300mm. Now we focus on an object about 20 meters distance. To achieve focus the lens to film/sensor distance is elongated about 2mm. It is possible that the camera software is reporting the back focus distance as 302mm.
It also depends on sensor size. Nearly all focal lengths are given with old 35mm values in mind (the exception being mediums format and larger). As an example if you see a 70-300mm Nikkor FX lens on a D7200, you'll get an effective focal length of 105-450mm (all values multiplied by 1.5). The effective recorded focal length might not make sense based on the lens' stated focal lengths unless you check to make sure of the camera in use was 35mm equivalent or APS-C, just as an example.