Translation:You are not going to be able to forget me.
Could anyone explain what exactly is moci here? Is this a form of the noun moc? If so, is být moci an expression? The structure of this sentence may make sense to a native English speaker, but being a native speaker of another Slavic language myself, I can't wrap my head around this sentence...
Oh, OK, got it, thanks. I'm a bit surprised that you use the form moci in the lesson instead of moct. I'm not a native Czech speaker, but I think ending some verbs with -ci and not -ct in the infinitive is generally considered to be archaic, no?
For example, the Czech Wiktionary page for moci states the following: zastaralý a knižní tvar infinitivu slovesa moct.
This is the imperfective future tense, it simply uses a form of "budu" plus the infinitive (e.g. "budu dělat"). When negated, only the auxilliary "budu" is negated (e.g. "nebudu dělat").
The double negative certainly does not apply to more than one subsequent verbs. For example "I don't want to see him" is "Nechci ho vidět". Saying "Nechci ho nevidět" cancels out the negatives just like "I don't want to not see him" would.
Fact is, from the point of view of a Czech speaker, there ARE NO double negatives in Czech. Words like "nic", "nikdo" or "nikdy" simply go with negative verbs and with each other. It's all part of a single negative to a Czech mind. A double negative is that "nechci ho nevidět" example, which works the same as in English.
"Nebudeš nemoci zapomenout" is a double negative and a weird way of saying "Budeš moci zapomenout" (you will be able to forget). We can even do a triple negative: "Nebudeš nemoci nezapomenout", which breaks the brain, but ultimately means the same as a single negative, similarly to "you won't be unable to not forget me". (Why didn't you ask why it's not "nezapomenout"? :D)