There were 16 French monarchs all named Louis, how was this tradition started? Has any royal line surpassed the French in repetitive names?

Louis is the French form of the Frankish name that is usually transliterated into English as Clovis, but was presumably originally something like Chlodowig. Other names derived from the same source include Lugvig, Lewis, Luigi and many others. Clovis I was the first king of the Franks to convert to Christianity and a major figure during the early medieval period (or dark age if you want to ruffle some feathers) in European history. The nations of France and Germany are both descended from his realm after it was split up amongst Charlemagne's grandsons in the mid 800s, so it's no surprise his name has lived on in various forms. If you're at all interested in the topic, then the episode Thor's Angels in Dan Carlin's Hardcore History focuses on Clovis quite a lot.

Sorry for not having a better source than Wikipedia, but I don't think this question is advanced enough to need one.

For the second part of your question, I can't think of a royal name that has been used more than 16 times, but might have forgotten or never heard of an obscure one. The Scandinavian kingdoms of Sweden, Norway and Denmark do have some really high regnal numbers, but they're widely considered to include several of mythical kings who never really existed. Of course if you count the popes as royalty, then there have been 21 named John and six other names used more than a dozen times!

Oh, just before finishing this comment I learned about Heinrich LXXII of Reus and Ebersdorf. I assume his regnal number was calculated differently to most, since even with a family that had kids as early as possible and absolutely no imagination when it came to naming them, 72 generations would place the first well within antiquity. If I can just make a follow-up question to my own answer, then I'd really like someone who's better versed in German aristocratic matters to explain exactly what's up with this guy.

edit: The case of Henry 72 is actually clearly explained in the Wikipedia page that I linked to. Their family has a habit of numbering all males, even if they never end up ruling. Not entirely sure how I manage to miss that.