Good news if you have three kids or more under the age of 18. The French state is there to support you, even if you are but a resident foreigner. The carte familles nombreuses entitles you to discounts from some 90 commercial partners, the most significant being the SNCF, the French railroad. Families with three kids are entitled to a discount of 30 percent on train travel with the card, with progressively larger discounts for bigger families. There is a 19 euro fee for processing the dossier; once received the card is good for three years.
You may request the card on-line and enrollment materials will be sent to you by mail, including all the details about the supporting documentation you must supply and a postage-paid return envelope. Why waste time? You can get started right now.
FAQ from the Ministère du Travail, de la Solidarité et de la Fonction Publique (in French)
Details on train travel discounts from the SNCF (in French)