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Are there any languages with strong polysynthetic tendencies that also make use of auxiliary verbs? It seems unlikely, but I'm no expert.

You have to define where you want to draw the line between polysynthetic and non-polysynthetic languages. I don't remember seeing any auxiliaries in the most morpheme per word heavy languages I've taken a look at but there are languages still pretty high in the synthesis scale that do have auxiliaries. The Salish languages are a good place to look. Here's a glossed example from Halkomelem (from Kroeber's "The Salish Language Family" with slightly simplified transcription),

ʔi=cən

AUX-SG1.S

ʔaa-t

call-TR

kʷθə=daktə

ART-doctor

k'ʷə=s-ƛ'emə-θ-am'š-s

ART-NZ-look-TR-SG1.O-3.POSS

"I called the doctor to look over me."

I don't know if it's telling or not, but here only the monimalised complement clause k'ʷəsƛ'eməθam'šs has a high morpheme to word ration while the finite part having the auxiliary construction has only two morphemes per phonological word.

I'm gonna guess that the impression you are getting that polylangs don't have auxiliaries is based on languages of the Eskimo type, which really aren't representative. Auxiliary verbs aren't uncommon at all in polysynthetic languages-- they just tend not to be the sort of "auxiliary verbs" that you're maybe used to, where the inflected auxiliary encodes modal and aspectual information of the non-finite main verb. Instead, the auxiliary serves as a second head to slot on additional (or sometimes pleonastic) morphemes in addition to on the "main verb". I can't bring out any examples right now, but I recommend looking up Osage, Jarawara, and (iirc) Ngan'gityemerri's systems of verbal morphology.

Yeah, I was going to say that Osage has polypersonal inflection and still makes use of auxiliary verbs. (Or not, depending on your point of view; since multiple inflection points are commonplace in the Osage verb complex, Quintero admits that the decision to separate auxiliaries from the main verb is somewhat arbitrary.)

Here's an example: ąmąbrį alee íbrilą ątxąhai
Wa-mą-Wa-ðį Wa-kik-ðee i-Wa-ðilą ątxąhe ðe
1SG.A-PREV-1SG.A-walk 1SG.A-SUUS-go.there INS-1SG.A-think 1SG.A.stand DECL
"I think I'll walk home."

As you can see, the first verb of the complex, mąðį, has two points of inflection. So there's a case for considering ątxąhe (which probably contains a fossilised from of the 1SG.A inflection) a progressive/intentional suffix to íbrilą rather than a distinct auxiliary, but keeping it distinct makes the predicate as a whole easier to parse.

I'm probably guilty of Eskimo bias, yeah.

From a paper I found titled "Covert Tense in Jarawara":

Ohi nenano amaka.
ohi na-hina-no ama-ka
cry AUX-IRR+M-IP.N+M SEC-DECL+M ‘He almost cried.’

So in Jarawara at least, the auxiliary appears to be more of something to attach morphemes onto in situations where the main verb can't take them, rather than something that in itself adds to the information being given. Is that an accurate way of characterising that?

Natchez has polypersonal agreement and complex verbal morphology and uses obligatory auxiliary verbs - I think this is also common in Muskogean languages. Basque of course does it too (though its claim to polysynthesis is probably less convincing).

Radagast revived wrote:I think this is also common in Muskogean languages.

Chocktaw has polypersonal agreement (including agreement for recipients) and it uses lexical auxiliaries to express a variety of modal and aspectual functions as in

Apa-t

eat-PART

tahli-li-tok.

complete-SG1-PST

I ate it up.

Still, its general level of synthesis strikes me as only medium high and I don't know if it's enough to call the language fully polysynthetic, whatever we choose to understand that to mean.