I remember reading somewhere that English has 14 vowels (presumably
including diphthongs), but that every dialect collapses at least two of
them together.

Well, I decided to measure my own lect, and got some surprising results. I
have at least 17 vowels that I can think of, all of which can appear
between /h/ and /d/...

/hid/ heed
/hId/ hid
/hed/ head
/hEd/ haired
/had/ had
/hAd/ hard
/hVd/ HUD - a bit of a cheat, since it's a acronym
/hOd/ hoard (or whored)
/h@U)d/ hoed (as in the garden tool)
/hUd/ hood
/hud/ who'd
/h3d/ heard
/hI@)d/ *heared (also seen in "beard")
/hAj)d/ hide
/hAj@))d/ hired
/hej)d/ heyed (past of "to exclaim 'hey'")
/hOj)d/ *hoyed (putative past of "to exclaim 'hoy'", which is a real
exclamation, but I'm not sure I've ever seen it verbed).

There may well be some missing /:/s in there, as I'm pretty bad at
detecting vowel length in my own speech. The /Aj)/s might actually be
/Vj)/ in my lect, or something in between.

More than just something for y'all to chew on, since I'm still kinda half
working on that onset/peak/coda writing system for English, and I think I
need some opinions on what I've managed to drag together so far. I might
need a distinct /ju/ from /u/, but I can only think of one case: /dud/
dude ~ /djud/ dewed.

Anything obvious I've missed?