How to export own objects using expdp without special privileges?
Creating Directory Objects and Export/Import are two different operations. Directory Objects are owned by SYS. 


  A directory object maps a name
  to a directory path on the file system. DBAs must ensure that only approved users are
  allowed access to the directory object associated with the directory path.
  If you are not a privileged user, then before you can run Data Pump Export or Data
  Pump Import, a directory object must be created by a database administrator (DBA) or
  by any user with the CREATE ANY DIRECTORY privilege.
  
  Schema Mode
  
  A schema export is specified using the SCHEMAS parameter. This is the default export
  mode. If you have the DATAPUMP_EXP_FULL_DATABASE role, then you can specify a list of
  schemas, optionally including the schema definitions themselves and also system
  privilege grants to those schemas. If you do not have the DATAPUMP_EXP_FULL_
  DATABASE role, then you can export only your own schema.


Data Pump Export-Schema Mode