The Tangraphic Sea rhyme dictionary for the extinct Tangut language distinguishes 105 different rhymes, disregarding tone. Tangut is generally thought not to have had final consonants, so how is this possible?
First, there’s a difference between counting of rhymes and counting of vowels. Finnish is often said to have eight vowels—ä a e ö o i y u—but it has 34 medials: eight vowels, each with a length contrast, and 16 diphthongs. Estonian, a related language, has nine vowels—ä a e ö õ o i ü u—but 63 medials. (More than that if diphthongs contrast for length.)
Rhymes technically include the medial and the final, as well as any semivowel that may occur before the medial, but if Tangut had no finals, its rhymes can be said to be equal to its medials. Estonian has 63 medials, but 63 is still far from 105.
Tangut may have been a Qiangic language. Southern Qiang has a large inventory of medials, and it may not have had any finals until recently. (Though the set of permissible single initial consonants is not identical to the set of permissible single final consonants, the set of permissible initial clusters is identical to the set of permissible final clusters: hlhieah-bburr ‘loess soil’ and cheahlh ‘sip’, hhsse ‘hot, numb’ and lehhss ‘book’, vshe ‘manure’ and wevsh ‘horse dung’ (from earlier *we-vshe; we is ‘horse’, and final clusters developed through loss of final vowels.)
How many medials does Southern Qiang have? There are eight vowels—ae a ea e o i iu u—and all but e contrast for length. Word-initial high vowels may be nasalized, and e appears nasalized in the affirmative reply ẽhẽ, which has pretty much the same pronunciation in English. Then there are fifteen native diphthongs (one of which only appears as a result of person marking) and one native triphthong, and two diphthongs and two triphthongs that only appear in Chinese loanwords. The LaPolla & Huang grammar isn’t clear on whether diphthongs and triphthongs contrast for length, but I don’t see any examples of long triphthongs, so I’ll assume they don’t exist.
There’s also contrastive r-coloring on most vowels; this is reconstructed for Tangut, though the sources in Tangut (preinitial, initial, and coda r) and Southern Qiang (medial and final retroflexes and coda consonants in some Chinese loans) are different.
Seven vowels and seventeen diphthongs that contrast for length, plus one vowel and two triphthongs that don’t: that makes 51. The grammar says that “all vowels can take r-coloring when they are the final vowel of a verb with first person plural marking”, but gives no examples of r-coloring on triphthongs or on diphthongs ending in -i (though ir is one of the four r-colored vowels that can appear in lexical items, along with ear er ar); if we assume that ‘all’ means ‘all’, that means Southern Qiang has 102 medials, not counting the nasal vowels.
Some of the Tangut rhymes were in complementary distribution and did not contrast (many, but not all, Class 3 rhymes only occurred after v-, l-, and alveolopalatals); Marc Miyake reconstructs 95 contrastive vowels.
However, Miyake’s Tangut reconstruction has more monophthongal places of articulation than Southern Qiang: æ a ɛ ʌ ɔ e ə o ɪ ʊ i, and then əu in place of monophthongal /u/. But this is not so different than English, which has /æ ɑ ɛ ʌ ɔ e o ɪ ʊ i u/—though many of these vowels are diphthongized in many dialects. (Mine has a vowel inventory that’s something like [æ~eə a~ʌi ɑ ɛ ɔə ɛi əu~o ɪ ʊ ɪi ɵu~u æɞ oi], leaving out the rhotic diphthongs.)
But twelve places of articulation is pretty much what we see in the Germanic languages, if we leave out the Ingvaeonic languages (English, Dutch, Frisian) and their diphthongization of close-mid vowels. Kensiu has 14; I think that’s the record.