How to export own objects using expdp without special privileges? As per this post on Oracle forums expdp can be used to export your own schema without any special privileges:


  It all depends on what you want that user to be able to export/import.
  If all that use is exporting/importing is their own objects, then they
  don't need any raised privs. If you want that user to be able to
  export objects owned by other schemas, then they would need the
  exp_full_database and in order to import, they would need
  imp_full_database.


However I just can't get it to work. Directory object is required to be specified otherwise expdp displays error:
ORA-39145: directory object parameter must be specified and non-null
But creating directories requires CREATE ANY DIRECTORY system privilege.

How then can I export my own schema as a normal user?
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