This wasn’t a game but it was one of my favorite activities when I was about 4. My mom would get out a selection of magazines, and we would make a collage. The first few times we did it (once a month or so, I think), she would just let me cut out whatever I liked and glue it down. Then we would have specific things to look for like, animals, people’s faces, colors, etc… It was a good way to learn a lot of things that was basically free and entertaining. It helped me with hand eye coordination, as well as learning a variety of useful things like how to identify emotions from faces with words or different types of animals (mammals, reptiles, etc…).
Play-acting/dress up games were big around our house. My brother had an assortment of costumes and would dress up in whatever and act it out.He was a bit peculiar and had a clip-on tie to play newsanchor. I would dress up in a little apron or something to play house, or we would dress up like people in certain trades and act out their jobs.
The other game we played often was scrabble. Of course I didn’t know how to spell much, but we would try to make the biggest crossword we could just by picking words and names and spelling them out.
I wish I knew more up-to-date games but my mom played a lot of games with us that she made up or didn’t cost anything.