What are the best non-government themed/non-Smithsonian attractions?

Like someone already mentioned above, Great Falls National Park is great for climbing, hiking or just spending the day chilling out and having a picnic. 

Also Mount Pleasant area just North of Adams Morgan is worth a visit. It has pleantly of nice cafe's and restaurants and isn't packed with tourists. Focus on Mt Pleasant St. Right next to it you have Columbia Heights which offers the same kind of vibe but even more people.

Rock Creek Park is a great place to run, hike, bike (Beach Drive on weekends is closed for motor vehicles) or just hang out if you don't want to leave the city. 

There's a lot of slacklining in Meridian Park between 15th & 16th St (just above V St) on the weekends if that's up your alley. Also every Sunday the same park has a cool drum ring and people dance and have in general a great time.

And, if you enjoy beer, google micro brewery tour in DC. 

You should check out The Kreeger Museum!  It is a private residence that opens up for a small group of people to do a tour.  It focuses on one to two artists and does a very in-depth look at their work.

I'm also a big fan of The Phillips Collection

A trip to  Great Falls National Park is also a nice summer activity in DC.

People seem to like the Newseum and the International Spy Museum. But I don't think that's what you're asking about.

LOVE the Washington National Cathedral.

The Fdr Memorial, Washington at night is a must-see, IMHO.

Ben's Chili Bowl and Jaleo are two great DC restaurants that you should visit. 

See some shows at The John F. Kennedy Center for the Performing Arts or Arena Stage.

(Hope I helped!)

I would totally recommend the International Spy Museum right after having an incredible pizza Ella's Wood-Fired Pizza on F Street right down from the museum.

http://washington.org/  One of my favorite websites!

Also there's the National Geographic Museum and International Spy Museum 

Totally depends on what you like to do.

For museums, I second The Phillips Collection which is in the embassy row area near Dupont Circle. It's a cool area anyway with a lot of great eateries. I like BGR the Burger Joint, but there's lots to choose from.

Go to Georgetown (Washington DC). You can see the Old Stone House which is the oldest building in DC (free and takes 5 minutes to tour), shop, eat, and do a chocolate tour. Walk to the Key Bridge and check out the Key Bridge Boathouse (http://keybridgeboathouse.com/) to rent a kayak and paddle around the Potomac. Before you go, check out the  The Exorcist Steps famous for the movie filming there.

Just north of Georgetown is the Washington National Cathedral. It's absolutely gorgeous and has a window with a real piece of the moon in it. There is both parking and a metro stop nearby. I'd also rent a bike and bike through Rock Creek Park (www.nps.gov/rocr/).

Take the metro to  Arlington (Virginia) and see the Arlington National Cemetery if you haven't already. 

Take the metro to Alexandria (Virginia) and spend a day there. You can also catch a water taxi from Alexandria to George Washington's Mount Vernon or the National Harbor. Alexandria itself is another amazing city with a lot to do and see, a lot of shopping, amazing restaurants, and some fun, smaller museums. Go to  Torpedo Factory Art Center to see amazing artists at work. If you like Lebanese, go to The Pita House on King Street.

For non-Smithsonian museums that are a lot of fun, go to the International Spy Museum and the National Museum of Crime & Punishment. The Newseum is interesting too.

In DC, I also love walking around Union Station just because it's pretty.

If you have a car, head to Annapolis or Baltimore for the day. Great Falls National Park is gorgeous for hiking, but could take 10 minutes if you just want to see it and leave.   Calvert County is exactly an hour away, but if you have the time and a car, there is a really nice hiking area with a beach where you can find Miocene Fossils - go to Flag Ponds Park and you'll find shark teeth, skate/ray dental plates, and tons of other fossils. Plus it's pretty.

Oh, and in DC, if you like history, go to President Lincoln Cottage. It's small, inexpensive, and by guided tour only, but it truly gives you an idea of what President Lincoln and his era of presidency were like.

The National Zoological Park, visit Historic Georgetown and walk the staircase from the Exorcist, Great Falls National Park, Visit the Washington National Cathedral and try to find the Darth Vader gargoyle, Washington Nationals game... and if you really want to have fun, come to Baltimore for a side trip!

If you haven't already, be sure to check out the Newseum  and the International Spy Museum in Gallery Place & Metro Center. Be sure to hike up to Brookland to visit the gorgeous cathedral as well. Have you seen the Washington National Cathedral? 

Rock Creek Park and Dupont Circle