Neatest rhymes with:
sweetest, advantest, anabaptist, anesthetist, apprenticed, artist, baptist, bluntest, brightest, cornettist, cultist, dentist, faintest, fastest, fattest, fittest, greatest, hottest, lastest, latest, librettist, lightest, noticed, orthodontist, practiced, quietest, quietist, remotest, slightest, smartest, softest, strictest, sweetest, swiftest, tightest, wettest
Neatest sounds like:
nantucket, natwest, neediest, netiquette, notched, noticed, nudged, nudist