Which famous (board) games are unlicenced and royalty free?
No singular registry exists. Nor, particularly, is it needed. Copyright expires after some 20-90 years, depending upon country and current laws.

So, anything older than 1920 or so is fair game. (Pun intentional.) But note also, if you simply reprint a work with an expired copyright, you can't copyright it in most jurisdictions.

The name, however, is another matter. Those are trademarks. US and UK Trademarks don't have a duration. Monopoly is, in fact, still a defended trademark, even tho' the rules have been in the public domain for years.  Trademark laws vary widely; consult a lawyer for details of yours.

Trademarks are readily searched using various search engines by country.

There is compelling legal theory (and a little US caselaw) that says you can duplicate the mechanics of a game without license, provided you reword it. You cannot, however, use the trademarks nor original text/wording, original art, nor overly close art. In any case, the line s fuzzy, and varies by country.