I live in the Logan Square/Avondale neighborhood of Chicago. Closest private I haven't played is Bryn Mawr, as I missed the GCA outing there and haven't otherwise made it over there. Closest public is probably the 9-Hole Waveland/Marovitz course, largely because I'm not generally an early riser, and 3.5 hours to play nine holes does not appeal to me.
More interesting for me would be looking at where I grew up in the West San Gabriel Valley in Southern California. I grew up in an unincorporated sliver of LA County between Arcadia, San Gabriel, and Temple City. Closest private would be San Gabriel Country Club, despite having driven by it approximately 1.34 million times. My mom grew up just across the street from the first green. My uncles used to sneak out and play 2-17, as they were all across the street from the piece of land that held the clubhouse, 1st, and 18th holes. Closest public was probably the 9 hole Scholl Canyon. No idea why I never made it over there. I worked at 2 different nine-holers a few miles away, so maybe I thought I didn't need to see a third.