Will Apple replace the broken Lightning cable for free

In general, extended warranty plans are a healthy-sized profit center for the company offering them. This implies that unless a product turns out to be less reliable than predicted, on average (mean average) customers will be better off paying for their own repairs without the warranty. However, if one can't afford to self-insure for abnormal or even normal deviations from the average, something like AppleCare will lessen the downside financial risks.

Thus, I suggest (and buy) AppleCare only if an Apple product seems to be getting numbers of reports of below average reliability (e.g. the Duo 2xx and PowerBook 5x00 product lines). For other Apple products, maybe put the money in a savings account instead, and there is a good likelyhood that you'll end up ahead after a decade or two of Apple products.

I found a discussion on this topic on the Apple forums. I tested a few of the solutions, including using a plastic collar stay, but the suggestion that worked was using two thin cardboard strips to release the grip. What worked for me:

1. Find a thin, semi-glossy business card, or similar piece of cardboard and cut it into two strips about the width of the USB port. You can use an un-occupied port to make sure the strip is narrow enough to fit.
2. Insert the strips on the two wide sides of the USB port, between the outside of the cable's plug and the inside of the USB port. You may need to work the strips back and forth to get them in. I had to insert the strips to about 2/3 the depth of the port. If you have difficulty inserting the cardboard, you may need a thinner material.
3. Grasp the sandwich of the cardboard strips and the USB plug. Pull. The plug should come out as easily as a normal plug would.