Is there a standard time limit, within which a car owner must report body damage to his insurance agent? IOW, if I have an accident, do I have to report the damage to my agent within X amount of time, after which they will not pay for the repairs?
I realize there are all sorts of policies, but is there a general answer to this question?
No, but it should be easily determined from your policy. Most policies these days are written in plain language, and an answer like that should not be difficult to determine; after all, the insurance company would prefer you know the rules than have to deal with disgruntled customers after the fact (well, they should prefer it, anyway…).
Call your agent if you have any question. You don’t have to say you had one; indeed, the way things are these days, I’d be quite certain NOT to say I had one unless I definitely want to report it, since even the mention of one can cause a rate increase in some states with some companies. 
As always it depends. As for a “general answer”, in my state there’s an “as soon as practicable” standard. It’s a generally a factual question (to be determined based on the evidence, case by case), so if someone can’t satisfactorily explain why it has not been practicable to give notice to the insurance company, the company could walk away. At the same time, if someone has waited clearly beyond the time when the insurance company could properly investigate the accident, a judge could decide it’s too late–for example courts have found eight months to be too long as a matter of law."
BUT, that’s just the general answer. As a practical matter, an individual is bound by the terms of the policy, and that’s where you have to look.
And an insurance policy is a pain to read, so it’s best to get your agent to show you the languiage.
I was hit by another car while my car was parked. Two weeks later I reported it to my insurance. They fixed it (minus my deductible), but when my policy was renewed, they jacked up the rates, claiming that I had had an ‘at fault’ accident. It turned out that since I had not reported the original accident to the police within 48 hours, I was assumed to be at fault. This appears to be standard for insurers in New Jersey.
Also, you can’t count different accidents as one repair, since your deductible applies to each of them.
My policy says “as soon as possible.”