"Not everybody is sleeping, but nobody is chatting."
Translation:Nem mindenki alszik, de senki nem beszélget.
I think that the Hungarian double negative works additive instead of multiplicative. i. e. "senki nem beszélget" does mean "no talking" + "it's also nobody who talks" rather than "(There's) nobody (who) does NOT talk". But thence I wonder how an actual Litotes like "He did not understand nothing" would be expressed in Hungarian.
The more negations you have in a clause, the stronger the negativity gets.
Disclaimer: I'm not a native Hungarian and still struggle somewhat with negations, so take the following with a grain of salt.
"He did not understand nothing" sounds to me more like "He didn't understand anything" with a quirky grammar that seems to be used in the Southern US, so if used with that meaning, I'd say "Semmit sem értett."
But on to what you had in mind: double negations. You could either be very smart and just use the un-negated sentence: "Mindent értett meg." Or you can use multiple clauses, a la "There was nothing (that) he did not understand": "Semmi sem volt, amit nem értett."