Level 3 unlocks the ability to request Cards from your Clan.
Level 4 will give you a single chance to change your name that can be done at any time afterwards in the Settings menu.
Level 8 unlocks Tournaments and 2 more slots to save Battle Decks to.
On 11/1/16, the Experience needed to reach level 9 from level 8 was increased by 25%.
King Level 13 was added in the May Update on 3/5/16.
Since the release of the Cannon Cart Card, if you upgrade all of your cards to max level, you can reach Level 13 with 107,791 Experience left over (assuming you do not collect any achievement awards and never donate).
If you were to upgrade all the cards to tournament standard instead, you will reach Level 10 with 4,591 Experience out of 30,000 to reach level 11.