What is the cheapest way to send books from USA to France? [closed]

The cheapest way to ship anything is to get somebody to carry it for you as a favour.

• You can ask a friend who you know is about to travel the route your stuff needs to travel.
• You can let all your friends know your predicament and ask them if they know anyone intending to do the trip in the future.
• You can check on social travel websites that you're already active on for people about to do the trip. CouchSurfing or one of its competitors for instance.

If the amount of stuff is small and/or the friend is a close/good one the price will be zilch.

If you're a nice person and especially if the mule was a friend of a friend or somebody you found on a social site then you could offer them a six pack of their favourite beer, a place to crash while they're in town, or even an amount of money you can afford but which is lower than the cost of using a shipping company would've been.

This is the cheapest way given no other restrictions.

If you have requirements such as needing the stuff within a short period of time, not having enough friends, or having so much stuff that part would need to be carried by each of several people, then that would be a whole other question.

Note that historically Travel.SE regards shipping that's not a requirement of your trip to be off-topic. Getting specialized gear to a place you'll need it for your adventure would be on-topic. (Climbing gear, hang-glider, off-road vehicle, kayak, etc.)

Note also that historically we tend to allow a lot of questions anyway if they're very specific so that they wouldn't get a whole bunch of correct answers or a whole bunch of guesses and speculative answers. In such cases you'll want to include all the information that's special about your predicament.