WordsI'm fairly certain that words that both alliterate and rhyme may exist, dependingbut a list of specific examples would depend on how you define "rhyme" and "alliterate". I think the most widely accepted examples would be things like blue, brew and cash, clash., where the words start and end the same way, but don't sound exactly the same.
If you use words that have the primary stress later than the first syllable, we run into some questions about the nature of alliteration. In Old English alliterative verse, apparently, the device of alliteration primarily consisted of matching the first elements of the onsets of certain stressed syllables. Words in the native vocabulary of Germanic languages tend to have stress on the first syllable, except for certain categories of prefixed words. Apparently, there has been some dispute about the importance of stress in Middle English alliterative poetry, but there is a case for it being closely related to stress.
As far as I know, the definition of "alliteration" as a device in contemporary English poetry is not so strict, but it does seem to be the case that matched onsets in stressed syllables are more prominent than matched onsets in unstressed syllables. (ThisThis is pointed out in "Alliteration, assonance and perception" from "Ling 131: Language & Style" at Lancaster UniversityUniversity; and I found a modern resouce "Linking Letters: A Poet's Guide to Alliterative Verse" that emphasizes a stress-based definition of alliteration.)
Remember thoughMy understanding (I still need to look it up in a source) is that by the previous definition of "perfect rhyme"in Old English, consonant clusters usually did not have to match entirely in order for the onsets ofwords to alliterate: only the primary-stressed syllables are not supposedfirst consonant had to match. So it is possible for a word starting with bl to alliterate with a word starting with br. The only class of exceptions that I have heard about is mentioned by tchrist in the comments below: consonant clusters of "s" + a plosive, like sp- and st-, did not alliterate with words starting with "s" followed by a vowel or another consonant in Old English.
So##Alliterative rhyme candidates
###Words starting with the same consonant, but different consonant clusters
Based on the above, it seems alliteration between different consonant clusters starting with the same consonant, but ending with different ones (likewould meet the requirements of both Germanic-style alliteration and perfect rhyme.
Examples: fee/free/flee,flee; pay/play/pray,pray; fight/flight/fright,fright; spy/spry,spry; die/dry) would meet the requirements of Germanic-style alliteration and perfect rhymedry; brew/blue; quick/click/kick; cute/coot.
The most effective kind###Rhyming words with alliterative 2ndary-stressed first syllables
If we tighten the definition of alliteration with entirely matchingto require entirely matching onsets, I think the strongest kind of alliteration that could occur between "perfect rhymes" in English would consist of pairs where one or both of the words have secondary stress on their first syllables.
##Rhyming words with alliterative 2ndary-stressed first syllables
Some examples:
##Rhyming###Rhyming words with unstressed 1st syllables that "alliterate"