There was a request here to show people answering questions when a bounty expires. The response was a bit unsatisfying as I think it is a reasonable request. While I think users can do without the expiration time, as someone who has created a bounty it seems very important for me to know this information. I have already had one bounty that was lost because I did not know the exact time it expired.
As opposed to the other request asking for browsers to see expiring time, I would like to see bounties show the exact time when they end at least from the bounty awarder's context. This would help us better award the full bounty and keep SO members happier.
You can mouseover the relative timestamp to get the exact time the bounty is going to end (like pretty much all relative times on the site):
In addition to Michael's tip (all times on our site support a literal mouseover tooltip, try it!)...
Bear in mind you also get:
So, in order to miss the bounty end, you have to try pretty hard.