I have a Prepaid American Express credit (gift) card of value $100. I have used it a few times and don't remember the remaining amount. Let's say I can't find out the remaining amount. Is it possible for me to go over the limit of the remaining amount when online shopping? Will the purchase automatically decline when over the limit, or will the purchase still go through? If the latter, how will I refill the difference due?
The fine print you agreed to when you used the gift card technically says that you agree to reimburse AMEX if they approve transactions that exceed the balance:
However, if a transaction occurs despite an insufficient Available Balance on the Card, you agree to reimburse us, upon request, for the amount of the negative balance created on the Card
That said, if it is a gift card as stated in the question, it would be highly unlikely to overdraw, especially with online shopping (and if for some reason it is, say based on a variance between what a restaurant pre-authorizes and what you end up actually tipping, the card issuer will probably eat -- no pun intended -- the discrepancy, assuming it is just a few dollars).
For additional reference, this article is about Visa in particular, but also mentions and applies to AMEX gift cards.
Generally, with a prepaid gift card any transaction for more than the remaining balance will be either outright declined or only approved for the amount that remains, depending on whether the system doing the purchase can handle a partial approval. It's much easier for the clerk at a grocery store to say "That only covered $50, there's still $2.31 left" than for a website to do so!
I expect that most online sites do not support partial approvals, which means that if you're trying to buy more than the card has left, you'll just get declined. If you can find out how much is on the card, which you can generally check online (for example, here for AmEx), you can turn any leftover balance into a custom Amazon giftcard which will let you pay for part of an order (on Amazon.com with this example, obviously) via what's left on your prepaid card and then the rest with a regular card.