morphological double negatives

Here's a question: under what conditions can one have a double-double negative prefix.  Here's (approximately) the example I read today:

I don't want a non-stick wok, I want a non-non-stick one.

Or what about this one?

There are only a variety of non nonprofit debt-consolidation companies

I'm wondering: what are the semantic and/or pragmatic factors that have to be present to get those double negatives? (I have my ideas...what are yours?)

This could be studied simply by collecting examples--from corpora, from the web, from real life.  For searching the web, I think one needs to start from a list of non-words (e.g. from a dictionary; and un-words, etc.) and then search with various negative prefixes in front--in double quotation marks on Google). There's a decent amount of literature on negative prefixes (that'd be where to start), but I don't know of any on stacking them like this. It's a rare phenomenon, but that just means that you might get to work from a nice small data set!

You'd probably need to look at which words take an extra negative and which ones don't, and have access to a larger context to try to figure out why the double negatives are used instead of a positive, where one would have been possible.

A related phenomenon is combining 'not' and a negative prefixed word--e.g. I'm not unhappy. That's been discussed more often (as it occurs more often).  Why?