A cloned one.
Seriously. The most expensive dogs are clones.

One American millionaire has apparently paid US $5 million to fund technology to get his pooch cloned.

Dogs were only successfully cloned for the first time in mid-2005. But it's a bit of a hot topic because it lacks the ethical problems of cloning other creatures (like people). And there are people willing to pay top whack ($millions) to get a carbon copy of Fido.

You can get a cloned kitty for a mere $50,000, by comparison.

Then there are well-trained guide dogs. An animal might be picked up at a shelter and then prove to be temperamentally well-suited to becoming a support dog (for blind people, the police, or people with special needs (like Down's Syndrome). It can cost close to US $100,000 to train such a dog, or trade it.

If you want to just talk about expensive breeds of dog (how boring!, anybody can have one of those!), the Americans pay most for English bulldogs, Thailand Ridgebacks, Egyptian Pharaoh Hounds and Lowchens.

Can't get firm statistics for Britain, but Petplanet reckon Shar Pei puppies here cost about £1000 each, compared to only about £400 for most pure-breds.

Or maybe you just meant which dogs are most expensive to keep. In which case, A) pedigree dogs, B) unhappy dogs, C) large dogs. Intelligent but under stimulated large pedigree dogs are the worst of all worlds.