Is there a possibility to be given a second chance after a inapte définitif?!??
It depends why you were given the inapte definitif. Your only recourse is to write to the general directly. You can write to him in English.
You must give the maximum info that you have such as your date of enlistment (when you signed your pre-contract) Your real name and your legion name. If you were given a matricule, (serial number) and your reason for leaving.
The address is the following:
Monsieur le général commandant la Légion étrangère
COMMANDEMENT DE LA LEGION ETRANGERE
Quartier Viénot - B.P. 21 355
13 784 AUBAGNE CEDEX
Writing to the general is a formality, he won't even see it. You should send the letter by registered mail with acknowledgement of receipt.
I hope that my making an exception to the rule will not open the flood gates to a load of questions from people who haven't even bothered to introduce themselves.