Hi Ben,
Add a text field to each of the three layers, with values "A", "B" and "C" respectively (use Field Calculator).
Do a Union of all the three layers.
Dissolve the union layer with the above three fields. The resulting layer will have 7 features with the above combination. Add a text field to the dissolve layer and define the desired values like "A", "B", "C", "A+B", "A+C", "B+C", "A+B+C".
Now if you want independent individual polygons in the feature class, you could use multipart to singlepart tool.
Think Location