Can any Nikon D5200 user tell me how many shots I can get on a D5200 with a fully charged battery?
A D5200 should be able to take about 500 shots per charge according to Nikon:
The D5200's superior energy-saving design allows you to shoot approx. 500 shots per charge* with the EN-EL14 Rechargeable Li-ion Battery.
*Based on CIPA Standards.
That number comes from an industry standard procedure for testing battery life of still cameras, and it's unlikely that your experience will exactly match the test conditions. Based on Romeo Ninov's comment below, we can modify Nikon's number and say that depending on how you use the camera, you should get 500 ±200 shots per charge.