It's been tried. A huge percentage of the internet runs on Amazon web services... And a massive ddos attack on that barely bumped it beyond the level of holiday shopping.
To get anywhere on "taking down the internet" they'd probably have to physically take out many sites across the globe.
Sea cables are probably the most vulnerable point of the internet. There are comparatively few of them (on the order of a few hundreds), they are long, and most of their length is not guarded at all. The only reason I can think of, why nobody has targeted them at large is that it would also cut of the attacker.