On the other he inflicted The Family and the Arthur animated trilogy on us.
So really, while the trajectory is on the downward, it's still anybody's guess as to which way he'll swing next.
And this is a film that - if the trailer is to be believed - is about a woman given super powers by a new drug made by a posh Englishman working for the Taiwanese mob.
But - and I know you'll be shocked by this - the trailer and the film have precious little in common. It becomes clear very quickly that the marketing department had no more an idea than Besson as to what Lucy is actually about.
What they think it's about is Scarlett Johansson's character becoming a kick-ass action heroine and exacting revenge on those who wronged her.
What he thinks it's about is an essay on life, evolution and the human brain.
And in a way, they're both right. Which makes Lucy every bit as confusing as that sounds.
But let's start at the start.
Lucy is a good-time party girl who for reasons no one cares to explain is studying in Taiwan. What's she studying? Frack knows. Can't be important.
People must leave the States to go to college in Taiwan all the time.
However, a drawn-out attempt to not do a favour for a bloke she met in a nightclub ends with her being handcuffed to a briefcase and being forced to go into a hotel and deliver it to a Mr Jang.
I say forced. She could just bugger off with the case, seeing as blokie has already stuffed $500 (her share of the delivery 'fee') down her top.
But she doesn't.
As a result, he gets shot, she gets taken up to Mr Jang's room by more burly men than should reasonably fit in a lift, and is operated on - along with three unknown men - so she can smuggle a new drug into America.
So far, to be honest, so good.
As long as you ignore the inter cut scenes of a neanderthal woman from a billion years ago, and various wildlife clips that serve to do nothing more than re-explain the action for the hard of understanding.
Meanwhile, at a lecture in Paris, Morgan Freeman's Professor Norman is explaining his theory of how man is only using ten per cent of his/her brain and what could be achieved if said usage could be expanded.
Only not to 100 per cent. He has no idea what will happen if a human were to start using the entire brain.
But what of Lucy I hear you ask.
While the men who have been impregnated with the new synthetic CPH4 are put somewhere else, Lucy - for reasons that I'm sure were in no way contrived to provide an weak excuse to burst said bag of drugs - is chained up in a room on her own.
One botched sex assault later, she's having seven bells kicked out of her by a man who took the 'no' rather personally.
Actually it's nearer 10 bells, so severe and brutal is the beating we are "treated" to.
And so, her journey begins. The bag bursts and, after literally bouncing off the walls, Lucy finds she is able to operate guns, fight anybody who comes near her, and walk about without being challenged while carrying fire arms, drive, communicate to others via TVs, radios and phones, see the roots of trees, the physical embodiment of radio signals and diagnose health problems with just a hug.
She can also learn at record speeds, but you've seen Limitless so you knew that.
And all from having a man-made version of a pregnancy hormone pummelled into her system.
Still making sense?
Because from here, it gets super nuts.
That's not to say this film is not without merit.
Johansson's performance is everything you expect it to be - spot on, note perfect and totally believable. And it's only 90 minutes.
Freeman, though, just looks confused by the whole thing.
But then you would too if you start trying to work out how an entire, fully-armed, Taiwanese mob army not only manages to get to Paris (not actually knowing that's where Lucy was heading) but does so with enough artillery to keep the Russian army going for the next 20 years.
And don't start doing the maths about how quickly people get from Rome and Berlin to Paris. Even running at only ten per cent your brain will hurt.
Sure, the action scenes are handled well and the film looks great - but there is just too much being thrown at the screen.
It feels like Besson was going for a mash-up between 2001 and Limitless, but decided to make Taken 4 halfway through.
By the time the ending plays out you'll be digging out your eyeballs and stuffing them in your ears to spare you further pain.
Somewhere in here are several good ideas for films.
What we end up with, though, isn't one of them.