This has been posted on the forum previously, but I've only gotten around to finding the documentation today. It wasn't particularly easy to locate, so I'm posting the info here in case someone else finds it useful.
Citizens of a few countries (Australia, Canada, Chile, Israel, Japan, Malaysia, New Zealand, Singapore, South Korea and the US) can stay in Denmark for 3 months or 90 days (it varies by country) even if they have recently spent time elsewhere in the Schengen Zone. The details vary by country of origin; time spent in other Nordic countries is in some cases subtracted from the extra 3-month/90-day allowance. Citizens of the US and New Zealand have the best deal: Time they've spent in other Nordic countries doesn't reduce the extra time allowed in Denmark.
Anyone contemplating taking advantage of this policy should read the document to which I've linked very carefully. Also keep in mind that the additional Danish time counts as time in the Schengen Zone, so the traveler will be in violation of the Schengen limit if he/she subsequently travels to another Schengen country. He or she will need to find a way to return home at the end of the trip without entering another Schengen county. I have no clue what would happen with a change of planes in Paris/Frankfurt/Amsterdam/etc., but I don't intend to find out the hard way. If I decide to do this, I'll be flying home via a place like the UK, Ireland, Morocco or Turkey.
The following is a Danish-government website: