Came across the following quote:
Never try to outstubborn a cat.
Googling did not help. What does Outstubborn mean?
"Be more stubborn than" or "surpass or outdo in terms of stubbornness".
It's modeled on words like "outdo" or "outsmart".
It's not a standard word, per se, but it's a common construction. See the prefix "out-" in the dictionary.
Less common overall is the attaching of the prefix "out-" to an adjective ("stubborn"), or really anything other than a verb, which gives the construction a jocular or lowbrow quality (at least in a neologism), but the construction is understandable, and cf. "outsmart" and "outclass".
As for other adjectives used in this way, a Google search returns evidence for "outstupid", and other examples are undoubtedly available.
Your example sentence appears to have originated in Robert A. Heinlein's Time Enough for Love (1973), and the Corpus of Contemporary American English gives another instance of "outstubborn" in Linda Lael Miller's The Women of Primrose Creek (2002).
Just like outwit or outsmart M-W mean to to gain advantage over someone by being more smart, "outstubborn" would mean to be more stubborn than someone, to gain advantage by being more stubborn.