I realize this has been requested before, but people tend to miss a compelling point:
I've spent the past several weeks googling for an answer. I'm finally at the point where I've concluded that despite the vastness of the internet, the answer truly isn't anywhere on it. It's an obscure niche that I'm looking for the answer in. I ask the question. I realize it's a difficult question that no one knows the answer to, and it's going to require considerable amounts of effort for someone to come up with a solution. Thus I want to offer a bounty immediately.
These 48 hours are a waste of time. I have already exhausted what's on the internet, what I know, and what everyone I know knows. I know that without that bounty, the question isn't going to be answered.
Further, why do bounties expire? Maybe the person answering takes over a week to come up with that answer (as already stated, this is a difficult problem.) It's not just the question asker being punished (which, whether that's a good idea or not is an entirely separate question,) but the answerer is also being punished. Oh, sorry, your quality answer takes too long, the bounty is being taken away from you. My proposal, bounties should last until a quality answer is given (until the offerer gives the bounty, or accepts, or automatically after a certain time if an answer receives say 3 up votes). In fact, it should be possible to set up multiple bounties on a question.
I understand the reasoning behind the 48 hour wait (and I hope that the reason I've provided is the straw that breaks the camel's back - there's already many reasons against the wait period,) - but I've not seen a single good reason that bounties expire.