To follow up on Dan's comment, be sure you are adding the field to a Feature Class and not a Shapefile. See my screen shots below. The top one is the attribute table of a Shapefile, while the lower one is the attribute table of a Feature Class in a file geodatabse. They are the exact same data, but in different formats. Notice in the the Shapefile the unique ID field is called FID, where as the Feature Class calls it OBJECTID.
A Shapefile is actually stored on disc as series of inter-related files:
A file geodatabase is actually a 'special' directory on disc, and all the files that make up a feature class make no sense to us humans:
Boat loads of people use the terms Shapefile and Feature Class interchangeably, which is a pet peeve of mine....
That should just about do it....