If we're talking "before you started spending lots of money on headphones", then I could probably survive for a week with my Sansui SS-20 (something my stepfather had lying around and never used). It does need new earpads and is rather heavy, but it sounds decent.
But if we're talking "worst-sounding headphone in your collection", that means I have to suffer with the GE H-22 (which I got well after getting into Head-Fi) and its total lack of extension for an entire week, hoping my ears don't bleed. It's comfortable, but it'll take extreme EQ to make it sound even halfway decent.
That brings up a good point. I guess we can't put a restriction on EQ, or even modifications, in order to get through the week. That could actually be really interesting--if a headphone is so bad it compels its owner to redesign it just to try to make it usable, it just might spur a new DIY craze for a particular model. "Hey, I made a Sony V150 sound like an LCD-2! Just do x, y, and z, and deposit your disbelief in the nearest suspension-inducing receptacle" or some such.
Also, vintage headphones are fascinating, if for no other reason than some of the designs and interesting ideas (e.g. AKG's variable-distance drivers) that fell by the wayside as the general idea of what a "headphone" is evolved. I miss the innovation and build quality of that period (and I say that from the standpoint of somebody who was born in the modern era of headphones), though from what I understand even midrange headphones from today are usually significantly better than the majority of designs from back then.
I wonder if that means that today's crappy headphones are on the level of midrange vintage cans?