Denmark Government Open Data.
Open Government, Denmark API is a free Government API that requires no API key, is served over HTTPS and has unconfirmed CORS support.
Details from opendata.dk.

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The Open Government, Denmark API is provided by Open Data DK (opendata.dk).
Yes. The Open Government, Denmark API is listed in the Government category of the public-apis directory, which only includes APIs that are free or have a free tier. Check its documentation for rate limits and paid plans.
No. The Open Government, Denmark API requires no authentication, so you can start making requests without signing up.
CORS support for the Open Government, Denmark API is unknown. Test a request from the browser, and fall back to a server-side call if it is blocked.
Yes. The Open Government, Denmark API is served over HTTPS, so it's safe to use in production.