One of the professors at the University of Texas (where I was both an undergrad and a grad student in linguistics), Tony Woodbury, is an expert on Eskimo languages and did fieldwork on Cup'ik
in Chevak (I'd transcribe his pronunciation of that placename in English as [ˈtʃʰivæk]. He now works primarily on Chatino, which is spoken way down in Oaxaca, although FWIR he makes frequent references to Cup'ik anyway, even when discussing Chatino with the grad students he's advising
). Unfortunately, I don't think I have any of his examples of Cup'ik with me anymore, but maybe at least some of his papers can be found online.
I remember him making at least a few interesting observations about it.
One is that the numeral system is vigesimal (based on counting not only on the fingers but also on the toes).
Another is that it's so agglutinative that the only parts of speech that exist in the language, really, are nouns (e.g. arnaq /aʀnaq/ 'man'; I forget which syllable is stressed (the first?), but I do remember that the [ʀ] is a uvular tap as in other Eskimo languages), verbs, and a few particles (including uuminaqsaga [uːmiˈnaqsaɣa] 'darn it!' which is the only one I remember). Everything else is expressed just by adding suffixes. One example he gave of agglutination in this language was this word:
ivrucilistengqersugnaitut [ivʀuˈtʃilistə̃ŋqəʀsuɣnajtut] (which IIRC means 'they definitely don't have someone to make them waterskin boots')
formed from ivruci 'waterskin boots'. (I think the morphological breakdown is ivruci-li-ste-ngqer-sugnait-ut, but I don't really remember what the other morphemes mean
He also said that when you're pronouncing such long words, if you happen to stumble, you can't just pick up where you left off in the word; you have to start all the way at the beginning all over again. So you couldn't say "ivrucilistenga...-ngqersugnaitut" or even "ivruciliste...nqersugnaitut" or anything like that. Instead, you'd have to say something like "ivrucistenga- ivrucilistengqersugnaitut" (if you're lucky enough to only stumble once
Finally, Tony once told me an interesting anecdote about doing fieldwork in Chevak. One time, he was working with his informant, a kind of older guy, trying to get grammaticality judgments, so he formulated a sentence and asked the informant whether it was a "good" sentence. (Presumably, the sentence meant something like 'The dog was on the mountain.') The informant (apparently looking a bit bewildered) said something like: "...No, that's not a good sentence. Look around you. There are dogs everywhere here. On the roads, on our sleds..." So then Tony tried out another sentence, and then the informant said, "Ohhhh. A bear
on the mountain. Now that
would've been interesting!"