Stack Exchange has an excellent draft mechanism (ever seen the faint "draft saved" at the bottom-right of the input box?) that restores what you've said if something happens as you type your question and your browser loses the text.
This feature does indeed POST the form data to SE (for this post it went to https://meta.stackexchange.com/posts/0/editor-heartbeat/ask); it doesn't merely save to the browser's (potentially-volatile) localStorage or similar.
This periodic submission conveys two very interesting pieces of information/metadata: how long you've spent typing your question, and how long it is.
Using these metrics, it would be possible to construct high-quality analytics about users who write questions but do not submit them, presumably because the process of studying and articulating the question provides the necessary insight into the answer.
I know that I've gone to Stack Overflow: Your code psychologist a few times and found myself answering my own question in the process of writing it. (I'm asking this question on meta.SE because I think there's high potential for possibly-unintuitive site-wide relevance.)
Implementational/Post-MVP considerations
Obviously, this feature would need to be fully anonymized to be viable. The Submit button has not yet been clicked, the user has not agreed to publish the data. Some users may copy sensitive information into the input box and then edit it out (remember to comply with PCI-DSS, yall!).
The good news is that I think this kind of feature would work just fine with heavily-anonymized data, and the statistics produced would still be extremely interesting.
The most basic implementation would merely look at time taken and form length.
Using something like Google's "diff-match-patch" algorithm to analyze text added, deleted, and moved around over time, it would also be possible to derive insight from exactly what point in time (banging the question out, revewing it, revising code, etc) the magic lightbulb moment happens.
Adding specific analytics to the question page itself would produce the highest-quality results. The most obvious idea would be to more accurately track server interaction so the point at which the tab gets closed can be identified with high precision. Another good idea might be to analyze when the tab is focused or unfocused, and maybe even do averaged keypress timestamping (eg, "started typing at T" / "stopped typing at T+N" / "resumed again" etc).
(On a related note - I see a fkey parameter in the submission, but I don't know if this can be tied back to the load timestamp of the question page itself. FWIW, from a security standpoint, adding a "load_timestamp" parameter to the heartbeat would make things a little gameable.)
Finally, the most interesting thing that could be implemented is doing keyword analysis on the text. You've probably seen SE's tag suggestion engine for unsubmitted questions (mine were timestamps markdown draft data-explorer user-interface search careers), so keyword analysis is already being done. That tag suggestion system could be tied into this new bounce-rate analyser idea to categorize unsubmitted questions by tag!
One thing that might be a good idea for everyone to do is highlight how comfortable they would be with the above analytics happening. Data collection is always a tricky topic, even when it's anonymous; some concrete unambiguity about everyone being okay with it can't hurt.
I also think that the phenomena of "questions people go to ask but never actually submit" doesn't consitute a risk to Stack Exchange itself in terms of the question not being submitted. I really do think publishing this kind of information would make for very good press. The site is still being loaded, the job ads are still being viewed.
PS. This question's title is 150 chars exactly. :D
Thanks to this forum thread for inspiration