Alright, for locals this is kind of like asking who's buried in Grant's Tomb, but it's a legitimate question, or at least one I'm asked fairly often. And in all fairness, those of us who live and work here should think hard about snickering at others. After all, just a few weeks ago, a friend of mine (and fellow tour guide), who we shall call "Mike", visited a large Midwestern city, perhaps Chicago, and was taken aback that their museums charged admission. In fairness to Mike, I too have an initial burst of surprise when I visit a museum outside the Beltway and have to reach for my wallet.
The short answer, of course, is that the Smithsonian is free, unless you count it's museum of design, the Cooper-Hewitt Museum in New York, which will set you back fifteen bucks. Oh, and the Udvar-Hazy Air and Space Museum out near Dulles International Airport will charge you fifteen bucks for parking, so that is a de facto admission charge as it almost, but not quite, impossible to get there any other way.
But for all the Smithsonian Museums in DC, and specifically on the National Mall, the cost is nothing. That's right, they're free. And not just them, the National Gallery of Art (also on the Mall), the Building Museum, and the Holocaust Memorial Museum, among others, also charge no admission
Now, I'm sure all you Heinlein fans out there are crying TANSTAAFL, and you're right. The truth is, you've already paid your admission to these Museums, whether you choose to read every exhibit placard or sit at home eating Cheetos, assuming, unlike some of our City officials, you actually pay your taxes. The Smithsonian is operated as a trust by Act of Congress and acts, more or less, as a Federal entity. Something like 70% of it's operating budget comes from the US Treasury, with the rest made up of gifts, contributions, and other proceeds such as that eight dollar hamburger you bought.
So what does this all mean? It means that you're being shortchanged if you don't visit. This may be the most direct return you get on your tax return, so take advantage of it.