Proto-Germanic reflex

How did Proto-Germanic *furi come from Proto-Indo-European *per-?

It can't have come directly from it. At one point, it must have been derived from *fur by attaching -i to it. That's the only way to account for the -u-, unless the -u- was already present in the PIE word.

Would the -u- have come from the syllabic r?

Yes. That's why the -i must have been attached later. If it had been there in the original form, *fri- would have resulted because the syllabic r would become consonantal next to a vowel. Compare *fra-.

Ah, okay. Now I'm remembering a PIE word *pr̥h₃. Is this correct? What part of speech is it reconstructed to be?

I don't know, but it would become *fur in Germanic.

Ok, thanks.

Have you ever, for fun, hypothesized how the PIE phrase *déms pótis would have reflexed into Proto-Germanic? I tried applying the appropriate sound changes and eventually I wound up with Modern English **tivad/timfed/timbed.

*pótis has a Germanic descendant, *fadiz, so it's likely that that part of the compound would have remained recognisable. Seen as separate words, the two would have ended up as *timiz fadiz, assuming that the first word would have remained a consonant stem (which is somewhat unlikely). It's more likely, perhaps, that it had become a kind of cranberry morpheme attached to the still-recognisable second word, giving something like *timfadiz. That might have survived into Old English as *tīfede, giving modern *tived /ˈtaɪv(ə)d/. Alternatively, the first part might have been reanalysed as a vowel stem, giving *temafadiz or temufadiz. Either of those might have given the same outcome in Old English, but I don't know if the nasal spirant law would apply to a syncopated vowel in this way. In Old High German, the result would be *zimfat, modern *zimft or similar. In Old Norse, the result would be *timfaðr.

Would the second *-i- in *timiz have come in by analogy?

The idea of a cranberry morpheme is pretty interesting. That's what happened in Proto-Indo-Iranian, right?

I'm not very familiar with Old Norse. How do you think *timfaðr would have reflexed into Modern Norwegian, Danish, or Swedish?

Probably more or less the same, except with the usual loss of -r and respelling ð as d. Danish and Norwegian also often convert unstressed vowels to e.

Also, to see what a native term might look like, I calqued the Vulgar Latin phrase which led to the English word "square" into PIE, hypothesizing a *h₁eǵʰs kʷetwr̥mós, leading to Proto-Germanic *igswidwurmaz and finally to Modern English *igsorme /ɪgˈzo(ɹ)m/.