attractions that don't require advance purchase

Barcelona sights where you should not show up without a pre-purchased ticket: La Sagrada Familia, Parc Guell, Casa Mila/La Pedrera, Casa Batllo, Picasso Museum, Palau de la Musica Catalana (English tours can sell out). Possibly also Palau Guell, but that Gaudi sight is less often visited.

Madrid doesn't have the extreme crowding you'll encounter in Barcelona. Some folks have encountered enough of a ticket line at the Prado and the Palacio Real that they suggest getting those tickets in advance. They've recently changed the entrance procedures for the Monasterio de Las Descalzas. It used to be extremely difficult for independent travelers to get in. It's easier now, but I don't know how the capacity aligns with the demand. I'd stalk the website to see how early English tours are selling out. (The disadvantage of having so many people capable of speaking your language is that those folks are competing with you for space on English-language tours.)

Government-run museums may have some free hours each month. As of 2015, the Barcelona tourist office (there's a branch beneath Placa de Catalunya) had a handout with all that info on it. The Museu Nacional d'Art de Catalunya (MNAC) and the Picasso Museum were included. However, conventional wisdom has it that it's not smart to go to a popular museum during the free period, because it will be at its most crowded then. On the other hand, I don't see how the Picasso Museum could possibly be more crowded than it was when I visited it during a pay-to-enter period in 2015.

The Cathedral in Barcelona (not La Sagrada Familia) still has free hours during the mornimg, I believe.

It's free to see the exteriors of the many funky modernista buildings in Barcelona; you just have to track them down.

The Magic Fountain shows at the foot of Montjuic are free but I think they've been suspended due to the water shortage.

In Madrid, the Temple of Debod (genuine Egyptian temple) is free.

The Prado has periodic free hours; details are provided in this blog post, which mentions free times at many other Madrid museums. I haven't verified the accuracy of the information,

Thanks so much! Very Helpful

I should have mentioned that Barcelona does have sights where you can walk right up to the ticket counter. I've done that twice at the Sant Pau modernista site (as recently as 2019), and others have reported similar experiences. There are normally not significant lines at either the large MNAC or the smaller Miro Museum, both on Montjuic. There are many other museums in the city that aren't crowded, but visitors with only a few days do tend to focus on the places I've mentioned in this and my previous post.

Just a note that I visited Barcelona last month, and the Cathedral no longer has free entrance times in the morning or any other time of day, unfortunately.