Nothing (and this doesn't count), right? Oxford says no, anyway. And they're the authorities, right?
Try this out (say it out loud if you need to): what sound does the letter R make? Based on everything you know about consonants and vowels, wouldn't you call "rrrr" a vowel? After all, it acts just like a vowel. Doubt me? Compare the relative positioning your mouth assumes for R against A, E, I, O, or U. Bearing this in mind, as well as the subtle dialectical shifting of the "schwa" (especially that of my 3-year-old, which, in this case, more closely resembles the "soft" I; and what, really, defines the "perfect" (referring again to the Oxford from above) pronunciation of any vowel, vowels being, after all, the first phonemes to change in any living language?), I have now found a word, in English, that rhymes with ORANGE, at least if we presume that in order to rhyme with orange, the final consonant, as well as the previous two vowel sounds, must echo precisely that of the next word, then ...
... the word SYRINGE rhymes with ORANGE.