On a budget, Smithsonian National Air and Space Museum still earns its price.
Free (timed-entry pass)
Standing under the actual Wright Flyer and the Apollo 11 command module is a real thrill, especially for anyone who grew up on space. The catch: it's been in a multi-year phased renovation, so galleries close and reopen unpredictably, and free timed-entry passes are required at peak times.