If you have a wreck and pay out of pocket for the other driver's damages is it possible for them to take it and still make a claim and get more money?

Absolutely! Do not assume that just because they agreed to accept payment directly from you, that the incident is over. Insurance companies need to be notified of losses in a timely manner. If you don't report a loss and pay yourself and the other driver claims other damages, your insurance company may not pay for them if you did not report it. Injuries due to accidents can become apparent long after the accident occurred. Protect yourself by reporting the claim. That's why you pay insurance premiums in the first place.