This is probably a stupid question, but I'm just curious. Does 非常 express something to a greater degree than 很？ That's probably not the way to word it, but like let's say you use 非常好。Would that be "good" to a greater degree than 很好？
They are different, of course.
非常 literally means 'not common/uncommon', thus extraordinary ("out of ordinary, un-ordinary", just like 非常), exceptional, very, etc.
很 means 'very, quite'.
Usually, 非常 is stronger in the meaning than the simple 很. Just like you wrote, 非常好 is above 很好 in degree. One of the reasons perhaps is that 很 does not always mean very is that sometimes it is only inserted before an adjective to make it "predicate-capable". In such cases 很好 simply means "good", not "very good". This has just just been discussed a day ago. Check this question for details.
Also, there is a related question about answering to 你好吗？ with 很好。
The Chinese Grammar Wiki article Simple "noun + adjective" sentences writes:
Nouns are linked to adjectives with 很 (hěn).
In cases when 很 is used to link a noun with an adjective, it does not mean "very", it's just a linking word: a dummy-linking adverb.
...in the case of these "Noun + Adj." sentences, you just have to think of this usage of 很 (hěn) as an exception. It's just part of the structure.
The example they give is 他很高. Writing 他是高 to mean "he is tall" is incorrect, we need to say 他很高.
If you actually want to add the meaning of "very" into the sentence, you could use another adverb instead of 很 (hěn). One good choice is 非常 (fēicháng).
In other cases, 很 means "very":
In the construct 很...的, it also means "very". I think it's because the 很 is optional: compare 一个很高的水平 with 一个高的水平.
I believe repetition can result in 很 meaning "very", e.g., 他很高很高, but maybe only children that talk like this (like here).