What's the biggest number you can name or think of? And no cheating by picking infinity.
Of course any actual number made of digits can be made larger by adding one... so the real trick is to look at ways of understanding and describing vast numbers. Turns out there is a lot of mathematical notion that covers extremely large numbers.
Much of this notation and the concepts behind astronomically big numbers is in Who Can Name the Bigger Number? by Scott Aaronson.
Scott covers a lot of material. But hand on and take your time. How else will you learn about Busy Beaver numbers?