OK. So, how bad do you want this particular house? Are other similar ones readily available at similar prices? What have similar houses in the same neighborhood sold for recently?
If you can dispassionately say you can walk away and find another house you will like every bit as much, either withdraw your offer or just tell them your offer is your highest and best.
If you're about to get a screaming deal, and even bumping up your offer still makes it a screaming deal, then offer more.
If you'll just die if you don't get this particular house, offer more.
Realize a primary residence is not an investment. Its an expensive doo-dad. Its no different than a car. Buy the one that meets your needs and that you can afford.
I have at times responded to this "highest and best" request with a higher offer. I did actually end up buying a house in a situation like that, though after we had a contract we had appraisal issues and I got back to a better price. Much more often I've said "never mind". But for me its just an investment.
I do recommend you treat buying a house like buying a car. There's no such thing as a "dream car" unless you're talking about a Lamborghini or some such. They've probably made 10,000 copies of any particular car. If you're willing to walk out of the dealership without looking back, you're in a much better bargaining position. Unless you're talking about a villa on the cost of some beach resort, its no different with houses.