Travis points to a provocative question on Salon:
What 10 minutes of your life would you relive if you had the chance?
I’ve thought about this for a while, and it’s actually a very intimate, private sort of question. In many ways, it asks “what is your most cherished memory?”. And that’s a pretty incisive question, which is why I’m not sharing.
One important question is awareness: when you’re reliving your 10 minutes, are you aware of the future? Often we don’t recognize a meaningful moment until long after the fact. If you were aware of the future, then it might enrich or inform a particular moment (and you might not necessarily choose one that was happy).
Of course, if you did retain your present-day conciousness, you might aspire to a weird moment, like the moment of your birth. I imagine that would actually just be terrifying, but you’d be the only person who remembered their own first moments in the world. Or heck, why not relive 10 minutes in womb?
DarrenBarefoot.com 
There’s a science fiction story in which civilization ended up with people in a virtual environment, reliving the happiest moment of their life, having their memory reset, and reliving the happiest moment of their life, over and over again.
It is in a way the end-state of individual pleasure-seeking.
I have two small children. I’d relive my last 10 minutes of uninterrupted sleep. (Uh, I mean, I’d relive the first time I held them, five minutes each.)
I would have taken that to mean what ten minutes would you like to try over again.
There’s a Japanese movie named “Afterlife” that deals with this question very poignantly, except it’s not 600 second, it’s a single moment. The movie’s slow and a little windy, but it has some interesting things to say about the nature of the choice.
“Afterlife” – Karen beat me to it. It’s a great film.
Also, the Spielberg/Kubrick movie “AI”‘s much criticized ending dealt with this same idea.