You can select all of the feature classes under the Contents tab within ArcCatalog and it will give you a count at the lower left of all the feature classes selected. However, this will not include feature classes within feature datasets. To get a count of both you may need to result to a python script. I've attached one here that you can use. Browse to the toolbox in the Catalog window, double-click the script within the Toolbox, and then browse to the File Geodatabase. The script will report how many feature classes exist within the geodatabase.