This was once the standard spelling of the word height, comprising the adjective high and the same -th ending that was added to broad to give breadth and long to form length. But Northern speakers of English changed the -th sound to a -t; while heighth was preserved in southern dialects up to the 18th century, it was the Northern pronunciation that was ultimately adopted into the standard. The reason we don’t say breadt and lengt is that these are not common combinations of consonants in spoken English. Try saying them – it’s not easy! The spelling highth was preferred by Milton, which earned him the censure of Dr Johnson who, in the preface to his Dictionary, criticised the ‘zeal for analogy’ which led Milton to adopt this form.
The older form has survived longer in the US; judging from comments found in the online Urban Dictionary, heighth is particularly associated with southern California. One definition states ‘This is not a word, even though everyone in southern California uses it. Don’t combine width and height into heighth. That’s just wrong’. Another definition, which also links its use with southern California, warns that using this word makes you appear ‘about 30 IQ points dumber’. The heighth spelling clearly provokes strong feelings: there are no less than three Facebook groups dedicated to its suppression – one is simply called ‘Its Height not Heighth’ (notice the misspelling of it’s!). The Republican presidential candidate Rick Perry discovered this to his cost when he employed this form in a debate with Mitt Romney in Las Vegas in October 2011, leading to adverse publicity about his ‘pronunciation problem’. It’s a common spelling on discussion forums, especially ones concerned with DIY. In one poignant post on Yahoo answers a disappointed lover asks: ‘Why are women so shallow about heighth?’ Amidst all the well-meaning responses, no one pointed out the very real possibility that it is his spelling of height that is the real obstacle.