Three is the most hatches you can do per day, so that the market for pets doesn't crash. However, you can always buy hatches from other users who are selling them (you might see people advertise that they are selling hatches, this means that you send them incubated eggs, and they hatch them for you and you get to keep the pet, the price is usually 100k per hatch).
You can keep your pets on your profile, which will make them age and become teens then adults, but make sure you feed them or fill your food pen if you do that, so they don't die.
You can also put them in your showroom once you have created it, so that they don't age and don't need fed.
Or, if you are a rancher, you can put them in your rancher shop to sell them.
The battles aren't up yet, but they are being worked on.
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