Back in high school, when we got around to it (about once a year) the Neighborhood would publish an issue of the Menlo Math Magazine. It was sort of a random collection of real-world-relatedness, whimsy, and interesting problems. Not an academic publication by any standard. But that didn't stop us all from facetiously claiming an Erdős number of 6, when we heard that Mr. T's was 5.
I just found out that mine is actually 5, and this time it's damn near official, not tenuously based on a high school pamphlet. Over the summer of 2005 I worked on the Stanford ALL project. Mostly as unpaid labor, admittedly, but I did sit in on the meetings and I did point us to a couple of valuable data sources. This meant I got to interact with some big names, however briefly.
According to Language Log, the lowest Erdős number known for a linguist is 2, and Geoff Pullum's is 3. A quick Google Scholaring shows that Arnold Zwicky and Thomas Wasow, therefore, have Erdős numbers not greater than 4. And they were both involved in the ALL project, as was I, which makes my Erdős number not greater than 5.
...All right, none of us are actually authors of the paper in question. I appear in the acknowledgement footer on the first page, along with the other students involved in the project. But Zwicky and Wasow appear in the same footer, as does John Rickford (who doesn't have any papers coauthored with Pullum, at least on Google Scholar). (Well, OK, all those big names appear in the references, which I don't.) Their contributions to the project were certainly substantive enough to qualify criteria for what counts as `collaboration'. My contributions were far less substantive, but I still think they counted for something, and if I appear in the same acknowledgements (the same parenthesis in the acknowledgements, even), that's got to be worth something.
(Naturally, this all comes with the caveat that we've been counting non-strictly-mathematical publications, but that has plenty of precedent.)