Board Thread:General Discussion/@comment-26454561-20150609175643/@comment-5089372-20150609181514

No, it will not increase with each battle and it never drops down to a lvl 1 dragon except in tournaments. You will get random dragons to battle, you have a lvl 11 one now your next one may be lvl 10.

Teamlava has never revealed what non-tournament dragon level depends on but I believe your island lvl does play a small role. Really low lvl account (when you first have access to the Arena) do have lower lvl opponents. As your lvl gets higher you in-counter higher lvl opponents. I have never seen lvl 16-20 in regular battles though (I don't think I have anyway) and I'm lvl 150.