How to enroll to the course without asking for that course' teacher?

To join the course go to courses list first, expand the drop-down list on the course name and choose "send request" or "join" depending on which option is available. 
You will be taken to the group menu. Click on the learners group you want to join.

After you choose Actions->Join the Group you will be able to access all course content either immediately or after course administrator approval. User can not Join a group more then once.

Note: it will be possible to do only if members group is configured for that purposes by the course teacher.