Apart from the loan words mentioned here like Januar, allegro, Baldrian ... from the Golston&Wiese article or words like Scharlach or Abenteuer, there are a few groups of possible candidates. Also, disclaimer: I used DWDS for etymological information, I'm not a professional linguist, so please tell me if I got something wrong.
Irregular Schwa-syllables (monosyllabic roots)
Words that end on a standard schwa-syllable (-e, -er, -el, -en) are excluded by default, but there are some "irregular" versions as well.
- -em: Atem, Odem, Brodem
- -end: Abend (kinda related to Greek "epi"?), Tugend, Jugend
- Leumund: This is a really interesting one. I suggest googling it
Rare suffixes (monosyllabic roots)
German has a variety of suffixes (-keit, -ung, ...), but some words seem like they have a polysyllabic root because the ending sounds so distinct.
- -ig/ing: König, Honig (both words with an intensive history); Pfennig, Schilling (currencies); Reling (from Low German/Nautical language or smth)
- -ut/od/öd/at: Armut, Kleinod, Einöde, Heimat, Zierat
- -ich(t): Kranich, Habicht
Old composita (monosyllabic roots)
Most composita are really apparent, like "Baumhaus" or "Waldhorn". But there are some words which merged a long time ago and thus don't feel like composita, some actually look like having some irregular Ablaut.
- with all-: albern (all-wahr), allein (all-ein)
- numerals: hundert (zehn-reden) / tausend (schwellen-reden)
- with "Heim": Oheim (Onkel-Heim), Heirat (Heim-Rat, most probably)
- plant: Alraune (Alb-Rune)
Also, with "Amboss", it's prefix+root, but the second part died as a word so it may seem like one root. There are other words where the root died as a single word, but it's mostly with be- or ge- prefixes, so it's more obvious there.
Animal/Plant names (polysyllabic roots)
Now we get to the actual polysyllabic roots. The majority of polysyllabic roots I've found are names for animals or plants. My guess is they're old words which are rarely used but can't really die out (unless the animal/plant becomes extinct) so they don't change / get simplified that much. There are possibly even a lot more.
- Animals: Ameise (already mentioned here), Forelle, Hering, Uhu
- Plants: Ahorn, Efeu, Holunder (which was actually monosyllabic once?), Wacholder (see Holunder)
Actual polysyllabic roots
Apart from all of those above, I was left with 4 words:
- Morast: which was a Germanic word, was borrowed into French and then back into German, so it's not 100% hereditary.
- Brosam: Not being used anymore (at least I don't know it), but it's listed in dictionaries nonetheless
- Arbeit: (already mentioned here) which got its second syllable from once having been a verb according to some etymologic entries
- Monat: related to Mond (kinda obvious), but seemingly got a second syllable for being a word for a time period.