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This just means that they fill the octet for however many valence electrons they require without going over so if an element only has four valence electrons than this is "as far" as it can go towards completing the octet.
I think it is like a personification that states the fact that they will share in covalent bonds or gain and lose electrons in ionic bonds to complete their valence electrons. If they did not "go as far as possible" many would remain incomplete and unstable.
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