This is why people often grant concessions just so the other side could save face and both could save the relationship. Naturally, this is often abused (by those who game the 'system'), but the point still stands: flat-our refusing is generally bad for you.
As much as we may all be tempted, there are better ways of responding (however less epic) than:
… without actually having to give them what they ask.
The short version is don't give it to them but still suffer through the process. The long version will take a bunch of paragraphs, so how about you go grab a coffee? This post isn't going anywhere and will still be here when you come back.
For starters, don't be extra stingy with your time. If they need five minutes or one or two additional e-mails to decompress, give them that. There's no need for you to put on a hard face and give them a hard time, either — not getting their way is already hard enough.
More importantly, it's always possible that the fine folks you negotiate with have enough authority or sway (you know the saying about the head and the neck) to see the deal followed through but still have to comply with certain policies and procedures without bending them more than they perhaps already are. Show them some understanding, and you'll have a friend — and chances are a mutually beneficial agreement every now and then.
Let's say they are required — thanks to the inifinite and indisputable wisdom of some policy-maker up there who has five doctorates but can't tie his shoes — to never accept the first offer but always negotiate, notably because many people will in fact give in at least an inch, so in the grand picture this will work for them, at least from the balance-sheet perspective (though perhaps failing to see the potential negative impact of aggressive panhandling on the company's goodwill in the long run).
So, if failure to negotiate a lower fee is the one thing they need in order to pay your fees and be done with the circus, then why not give them just that? And sit through the ordeal. It won't kill you, it'll make you stronger.
Just keep declining politely, expressing some compassion with their position and citing the existence, if not the details, of some objective factors influencing your decision or forcing your hand. Even if that's something like:
My fees are already on the lowest level I could realistically accept after lengthy negotiations. So, in essence, you're getting the whole benefit up front to save you and me some time. Unfortunately, this misses all the adrenaline.(Here, you rationalize how you can't go any lower, and you let them know that they are already getting the best possible outcome, hinting they shouldn't feel less satisfaction with it just because they didn't have to fight you for it. Mentioning that you respect their time is a gentle nod to tip the scales and make the refusal effectively not a refusal but an explanation hinting that perhaps they're getting more than they were asking for. In the end, the tension is resolved with comic relief.)
Depending on the situation you could add something to the effect of: 'and to treat all my clients equally and fairly whether or not they decide to negotiate,' but I'd be careful with that because a lot of companies want preferential treatment and favoured status (even MFN) regardless of not having done and not intending to do anything to deserve it. We do live in uniquely narcissistic times. For others, however, notions of equality and fairness will be of top importance.
And try to dissociate the poor souls from the policies and superiors that make them do things they wouldn't normally think about. In some cases it's a good idea to let the little people know you aren't blaming them for the big people's screw-ups. And even if the little people do screw up, show them some patience; we're all human after all. Especially in the little-people league that the bulk of us play in anyway.
Exception: Time wasters abound. It's good to learn to spot them, and most people eventually will, though it may take years. Even so, you generally don't lost that much time by simply reiterating your proposal and using more than two or three abrupt words to say that you won't be going any lower. Just don't get involved in a lengthy discussion rehashing and rebuffing the same old arguments, don't leave too many openings, and so on. Don't be chatty, but don't leave the table just yet. Just keep your time investment and especially your hopes down to the absolute minimum you won't regret spending even if nothing comes out of it — which is the almost certain outcome when your initial positions are too far apart.
Still, avoid unnecessarily alienating your contact. Even when the boss's budget ideas are patently unrealistic, the secretary or assistant will sooner or later end up working for someone who is more reasonable or fill the same position when the old incumbent goes up or out. Practical experience teaches that does indeed happen, and people remember their past contacts.