We have "best friends" and "close friends", because you give first importance to them. Like that, what should I call an enemy who is first enemy? I.e. you want to hurt him most then the rest of your enemies. What should we call them?
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As per Wikipedia, an archenemy, archfoe, archvillain, or nemesis is the principal enemy of someone or something. In vernacular English, archenemy and nemesis are the two from the four above that are actually used commonly.
There's no single English word to describe the single most important enemy in which case you need to combine two words:
Note to the OP: While archenemy has no friendly connotation, nemesis can also be used to refer to friendly rivalry.
You are probably referring to an arch-enemy (archenemy):
Examples of arch-enemy:
I suppose you could use sworn enemy per CALD: