Gebyr=Fee in Denmark

It should be well known that using a credit card is not free. The merchant pays a percentage to the bank, this fee can be as high as 4% IIRC. In the US, it is usually not allowed to advertise a discount for using cash, it would seem that the credit card fee is worked into prices in general, and we appear to be paying this fee in the USA whether we use a credit card or pay cash. Outside the US you can often find that there is a surcharge for using a credit card for paymen, or a listed discount for cash. It seems that the Gebyr in Denmark is the merchant adding on their fee for the use of plastic, rather than their having to put in in their price.

For that matter, this is the reason behind the hotels in Italy who insist that the local hotel taxes be paid in cash while paying the hotel bill with a credit card. they have to pay the full amount to the government, and do not want to have to lose the fee they would have to pay to the bank if the tax were paid via plastic..

That fee equals the amount the credit card issuing banks are skimming off the bill when the customer pays with a credit card. The merchant (hotel or restaurant in this case) has a choice to either raise their prices to include an amount which will offset this or simply charge it if they want the reason for the higher prices to be noticed. Either way, you will be paying it even if it is not noted in the small print. After all, someone has to pay for all the points or cash back everyone wants and it sure isn't going to be the card issuing bank.

I find cash always easier to use in most of Europe for most things.

Fees for foreign credit cards are the norm. The max allowed is 3.75%. However it is illegal for merchants to charge fees for Danish national debit card, the Dankort. With only a small percentage if cards actually having a fee, merchants cant include it in their prices.

I believe fees are disallowed for all EU cards.

Restaurants and hotels probably charge a fee since the amounts are quite big.
My experience before the EU regulation was that the more "touristy" place the less probability for a fee. Blue planet aquarium had a fee a then but many other tourist attractions didn't. Magasin and Illum didn't charge a fee, but otherwise fees was pretty common among supermarkets and retailers, at least outside the main shopping area. I never paid a fee for "street food".

My last Danish cash (probably ever, since cards are accepted everywhere and the number of cashless places is increasing) was withdrawn without fee from a DKK/EUR ATM at the airport's metro station since I the bank ATM in the city wanted a fixed withdrawal fee. Of course I had to take great care to answer the DCC-questions carefully. You probably won't find a currency exchange with less than 6% spread (corresponding to a 3% fee).

Avoiding fees completely will be a bit of a challenge.