Does Photoshop support 14-bit images?

Suicide  wrote I'd imagine that if you try to open a 14-bit photo in 16-bit mode, everything will look dark, since the photo's brightest white is only ¼ of the color space's brightest white.

That's not how it works, so the problem you describe won't happen. Bit depth doesn't describe a range, it describes how many steps you get within a range.

It's like having two thermometers that have a range of 1 degree to 100 degrees, but one thermometer is marked every 10 degrees and the other thermometer is marked every 5 degrees. Both can show you the complete range from 1-100 degrees, but the second one is more precise.

Opening a 14-bit raw image as a 16 bits per channel document doesn't lose any quality, because 16bpc has more "slots" to store the tones of the image, not less. It's like telling 14 people to board a 16-seat bus: Everyone gets a seat, nobody is left behind.

Also, only offering 8bpc or 16bpc is not a limitation of Photoshop, because it works the same way in other photo applications. 8bpc and 16bpc are standard bit depths for image files. 12 bits and 14 bits are used by cameras when storing raw data off of their 12- or 14-bit sensors (some high-end cameras can do 16 bits), but once you want to take that raw data and put it into a standard format, you will always end up in an 8bpc or 16bpc file in whatever photo application you choose.