This question directly affects me, since I became a "99er" last week… I can provide the following insights:
The US Unemployment Rate ONLY covers the following people:
Those unemployed receiving unemployment benefits, who must give weekly/biweekly updates to their state/local unemployment office to continue to receive benefits.
Those unemployed no longer receiving benefits, who volunteer their job search history/status on a regular basis.
In this sense, it's not just the discouraged who no longer show up – once you stop receiving an employment search form, and stop being incentivized to send it back (to receive your next check), then effectively the unemployment bureau is asking you to spend money on paper/printing and stamps to submit additional reports, and/or reporting physically to an unemployment office on a regular basis.
Neither case is particularly practical once the money has run out. It may be embarassing, but you really do eventually start measuring your life against the cost of a stamp… And ironically, the way the system is set up, it's exactly at this lowest of the low points, where you no longer are considered unemployed. Go figure.
The real rate, including unemployed, underemployed, and malemployed (overqualified workers who effectively take jobs outside their expertise/seniority level who need the work) is closer to, if not exceeding 20% in the US. And no politician wants to talk about rates at that level, which is why the current system of reporting was created to fudge the numbers/make them look better.
Since this system has also been in place for so long, both major political parties are loathe to use a more accurate count – because in doing so, they would effectively make it look like a huge percentage of jobs were lost on their watch.