Hi,
ArcGIS Explorer doesn't need administrative privileges at runtime - only during installation. As you observed, ArcGIS Explorer writes the cache to %temp%\E3. Additionally, other locations ArcGIS Explorer writes to are:
C:\Users\<USERNAME>\Documents\ArcGIS Explorer
C:\Users\<USERNAME>\AppData\Roaming\ESRI\arcgis explorer
C:\Users\<USERNAME>\AppData\Roaming\ESRI\ArcGISPackageCache
C:\Users\<USERNAME>\AppData\Roaming\ESRI\ErrorReports
Regards
Mike