Enjoy the Weekend Cultural Treats
Saturday and Sunday are free at the Nacional Centro de Arte Reina Sofía. The tiny Casa Museo de Lope de Vega is free on Saturday only. Sunday-morning-only treats include the Museos de América, Sorolla, and Nacional de Artes Decorativas.

Don't Forget the Weekday Freebies, too
El Prado is free in the evenings all week long (though it's closed on Mon). Some of the more individual private museums such as Lazaro Galdiano and Cerralbo are free on Wednesdays (also Sun for Cerralbo), while the Museo San Isidro in Plaza San Andrés gives you a daily rundown on the city's history for free. Also permanently free are the Bolsa (Stock Exchange), Museo de Historia, Museo Naval, and the Conde Duque cultural center.

Take an Ecclesiastical Trip Back in Time
I'm particularly fond of Madrid's two oldest churches, hidden in the heart of the Austrias district. They're both tiny, giving you an idea of what Madrid must have been like when it had a population of just around 10,000. San Nicolás de los Servitas (full title San Nicolás de Bari de los Servitas) was named after an Italian saint, and its 12th-century Arabic torre rises above a narrow lane just behind the Calle Mayor. Nearby San Pedro el Leal, also known as San Pedro el Viejo, has the best-preserved 14th-century Mudéjar brick tower in the capital. No entry fee for either, of course, though you may care to leave something in the collection box to help toward the preservation of these two gems.

Stroll in the Parks
Madrid's best central parques are particularly rewarding to explore, especially the Retiro, with its rose garden, fountains, statues (including the Angel Caído, or Fallen Angel, depicting Lucifer), and central lake; Casa de Vacas; and the 19th-century Palacios de Cristal and Velázquez cultural showrooms. Below the Palacio Real, the Campo del Moro has a verdant neatness more associated with Northern Europe, while on the edge of Argüelles the Parque del Oeste's marked nature trails wend their way down past an international selection of trees and plants to the River Manzanares (where you can view the Ermita de San Antonio de la Florida's Goyan frescoes). Most relaxing of all and only open weekends is El Capricho, first laid out in the 19th century and complete with the original fountains, gazebos, and waterways. It's on the eastern edge of the city just a 5-minute walk from the El Capricho metro station.

See Madrid's "Little Egypt"
On the ridge overlooking the Casa de Campo, you can visit the Templo de Debod, a unique slice of Egypt in Spain. It's the real McCoy: shipped stone by stone from the banks of the Nile.

Be a Politician for a Day (or Morning, anyway)
Visit the Congreso de Diputados on Saturday mornings, and imagine yourself changing the state of the Spanish nation. No need to book. Just turn up at 10am (except during the summer recess).