Challenging final offers is pretty difficult.
If the final offer?
a) the actual final offer and best available? or
b) a negotiating tactic to get you to accept less than the best offer available?
If it's option a then you're in danger of making it look like you don't believe the person telling you it's a final offer or if it's b then then potentially there was room for movement.
When final offers have been made, especially when going for a job then I've never challenged it, for two broad reasons:
a) generally you will end up working for the person you're negotiating with. Doubting their credibility isn't a good start!
b) secondly, I've completed my due diligence. I know what package I'm willing to move for and understand whether the company I'm interviewing with can meet my expectations. I will have made the interviewer aware of my expectations, there’s no point in going through a lengthy recruitment process if there is no chance of my package expectations being met.
When negotiating for one off purchases, where maintaining a relationship isn’t important, then I always push even when a ‘final’ offer is made.