As someone who might be more than a little embarrased to have someone she knows in 'real life' stumble on some of my past threads, I have to say asking to have them deleted is tempting. However, many, many people responded to some of my desperate pleas for help with long, thoughtful responses. I value the time they spent to help me out, so I would feel awful just making it vanish.
I am pretty much of the opinion that (except for situations like Erin pointed out) it should not be a possibility. It's sometimes painful when I come across an old post, but that just reminds me I have work left to do.
I have heard of people losing a job/not getting hired because of stuff on the internet (such as their MySpace account). But I keep a degree of anonymity by not posting my full name. Maybe one of the rules could state that deletion requests cannot be honored due to the workload and therefore do not post under your real name unless you intend to never say anything you might regret
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