What's the finest readily-available way to write on paper?
The smallest pen I have personally found is a 0.03 mm size pen; while it does leave a very small line, the tip is quite fragile and tends to break more often than my larger pens (including the 0.05, the next size up). It requires a delicate touch; while you would probably be fine with it, I wouldn't recommend handing it to kids, who haven't always developed the motor skills to be gentle with the pen.
The 0.03 size is also fairly uncommon; typically, the smallest pen tips you'll find will be either 0.05 or 0.1. You can see a comparison of the line sizes in this sample image from JetPens, found in their product listings for Copic Multiliners, one of the brands that does carry this size of pen; Marvy and Deleter also sell a pen in this size. The image below is an actual sample of the pens, on an undisclosed lettering paper, to show comparative line weight and bleed between sizes.