I can never remember how to work this out in Photoshop but I stumbled upon in Dropbox that for Nikon cameras at least it gives this to you.
However, you need an iPad (it might also work on Android). You select the image in Dropbox and at the bottom of the screen there is an i with a circle around it. Click that and you get a whole bunch of stuff. (It doesn't give you the shutter count in Dropbox on a Mac, however)
It has the standard information such as aperture, shutter speed and so on, but apart from the very useful shutter count you also get the camera's serial number.
For some reason though it doesn't give shutter count details for my Fujifilm XQ2. Only for Nikon bodies thus far.
As a side note, why do all cameras reset the file number back to 1 when it gets past 9999?
It's seriously annoying as I've had conflicts in LR if I forget to name my folders to say D800A then D800B and so on.