There are a lot of Foreign Legion Books available at Amazon et al.
The majority of them seem to me to have been written by guys who have done a single 5-year tour and recorded their experiences. They all read pretty much alike:
Fronted up at the gate;
Stripped of everything including dignity;
Shipped off to training garrison;
Turned into killing machine;
Joined 2 REP (Foreign Parachute Regiment: there is no 1 REP for historic reasons. Nobody seems to want to admit to joining any other arm of the Legion);
Went off to various war zones (many in Francophone North and Central Africa).
Simon Murray's "Legionnaire" (it has in recent years acquired a longer title) fits this description, but stands unique because it was the first first person account of the FFL to impinge on the public consciousness and also because it describes a turning point in the history of the Legion (including "What happened to 1 REP?).
It is far and away the best first-person Legion book: a must read, even if the events are from half a century ago. Pick one from many of the other first-person accounts and you will have the intervening years filled in. I have read several but none stands out.
I went through a phase of reading up on the Legion. I got a book entitled (IIRC - I pass it on the bookshelf every day) "Foreign Legion Parachutists": it has lots of pretty pictures and gives you an insight into the development of Legion parachute troops from 1 BEP in South East Asia through to the contemporary 2 REP, but it is badly translated from French and suffers for that.
Another good Legion book is the History of the French Foreign Legion: From 1831 to Present Day by Douglas Boyd, who seemed to have access to official archives, which gives it a degree of authority. My only criticism of this book is that I'd hoped to find out more about the de Gaulle Putsch of 1961. Unfortunately, the only information Boyd could find on the Putsch was the writings of Murray in Legionnaire. Still a pretty good book.