If you don’t want to use Python you can use Modell Builder. Model Builder has several iterator tools.

Use [Iterate Feature Selection][1] to iterate over features in a feature class (your watersheds).
Then use [Clip Tool][2].

You have to run this modell 10 times (one for each of your 10 raster).

Here are some [examples of using serveal iterate tools][4].


  [1]: http://resources.arcgis.com/en/help/main/10.1/index.html#/Iterate_Feature_Selection/00400000000s000000/
  [2]: http://resources.arcgis.com/en/help/main/10.1/index.html#/Clip/00170000009n000000/
  [3]: http://resources.arcgis.com/en/help/main/10.1/index.html#/Iterate_Rasters/00400000000z000000/
  [4]: http://resources.arcgis.com/en/help/main/10.1/index.html#/Examples_of_using_iterators_in_ModelBuilder/00400000001n000000/