Interesting -- this word (agheg) appeared in the "vocabulary" section in chapter 1 of my Western Armenian textbook, but I don't think it was ever used in a sentence: the book prefers the words pari and lav for the meaning "good".
How does agheg differ in meaning from pari and lav? Is it a stronger word, or perhaps more formal?
In reality, all three words have the same meaning. The only difference is that "agheg" and "lav" can be used as adjectives or adverbs, while "pari" is only used as adjective. In the present day usage, it is more common to see "agheg" used as an adverb than as an adjective, but if you use it as an adjective, it is not wrong.
"Lav" is used with equal frequency as an adverb and as an adjective.
Regarding "pari," even though it has all the range of meanings of "good," it is more common to see it used to express the goodness of a person in regards to his conduct in his relational behavior. For example, if you wanted to say "he is a good man" meaning that he is of good personal qualities and value, honesty, work ethics, and such things, that are judged in the person himself, you would use "lav," but if you were to talk about his being a good, compassionate, merciful person, in his relational behavior with other people, you would use "pari," the same as if you were to say that a child is well behaved and obedient.
Probably I didn't cover everything, but it is a start. And as with all the words that have more than one meaning and more than one synonym, there may always be nuances that come to us only with the use, reading and conversation.