Interesting idea of free museums. I can't think of one museum I've ever been to that did not require purchasing a ticket. Granted, in my limited travels abroad, it has been for work with no time for that kind of thing, so my experience is solely with museums in the US.

Many museums in the UK are nominally free (although they encourage a donation) and they charge for special exhibitions. A few in France are free. Can't speak to more broadly. (And, yes, free museums are pretty uncommon in the US although they exist--especially at universities, most of the Smithsonian Museums, and so forth.) Fairly broad experience even if it's often been in conjunction with work travel.

Have you never been to the Smithsonian in DC?