Dragon's breeding attempts are represented as eggs shown on the dragon's card. Each breeding attempt equals to one egg. When you run out of attempts, you can't breed this dragon with any other dragon.
You can give your dragon additional breeding attempt by feeding other dragons to it. This way those dragons are consumed, and the player loses them. You can only feed your dragon with dragons of the same rarity level or more common.
Each dragon has different amount of energy that can be refilled by other dragons. The stronger and rarer is the dragon, the more dragons he will need to be sated, and more energy it can refill to others.
Also players can use crystals to refill dragon’s breeding attempts.