First of all, walking into any of the major parks in Madrid will ensure a fantastic free-of-charge experience of nature and culture combined. They often host festivals, exhibitions, and even puppet shows open to the public for free. Secondly, you can always enjoy the most significant urban and shopping areas of the city, with their monumental architecture and public art, accessible to everyone. You can even enroll in one of the Free Madrid Tours, which last about 3 hours.
Also, most of the top museums have an entrance for free one day per week for several hours. For some, it is on Sunday, for others on Monday or after 18h. Check on their websites for the respective offers.
And last but not least, don't forget that most of the religious temples in the city are also entry-free.