This eight-year-old article from Scott Aaronson is thoughtful on many levels. Aaronson starts off innocently enough, with a "game" in which contestants must name the largest number they can in 15 seconds.
You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.
This exercise leads to a journey through human thought with unfortunate ramifications for our future.
One answer is to write down as many 9s as you can on the notecard. Of course, this answer would be shattered by anyone who wrote down a bunch of 9s raised to the power of a bunch more 9s. This is, in turn, blasted by anyone who writes 9^9^9^9^9^9, and so on. In fact, each of these improvements so far defeats the previous, that you could give your competitor some extra time, and still win. Instead of 15 seconds, you could give your opponent as long as the universe has so far existed, and you'd be plenty safe. Even 9^9^9 is vastly more than the number of particles in the known universe.
Aaronson goes on to show us that even these exponentials are child's play compared to strange entities such as the Ackermann series and, still crazier, something called "Busy Beaver" numbers that have to do the longest time a theoretical computer could work on a problem without working infinitely long.
All of this really gives perspective that the winner of this contest is the one who has the highest order paradigm. After you read about Busy Beaver numbers, you almost feel sorry for the poor saps who are still doing 9^9^9.
Aaronson also looks back through history, showing how far we've come. The bible said that pi = 3, and made numerous references to the number of stars and number of grains of sand on a beach being "infinite." That kind of thinking is what you'd expect from someone who just wrote three or four 9s on the notecard, lol. Archimedes came along and said the number of grains of sand must certainly be less than 10^63, therefore it is finite. Shocking stuff for his time. Incidentally, the church threatened death to Pythagoras of Samos for saying that the square root of 2 is an irrational number (its decimal form has infinitely many, non-repeating digits) and Socrates died for his scientific beliefs. Only in pockets of the world where reason had a chance to flourish did humans compute more and more digits of pi, and come up with higher-order paradigms of large numbers, and so on.
Why are people so afraid of large numbers? Why should anyone care in the first place?
Physics professor Albert Bartlett said, "The greatest shortcoming of the human race is our inability to understand the exponential function."
He's talking about the idea that if the human population continues to double every 40 years as it has so far, that the mass of all humans in the year 3750 would equal the mass of the Earth. Clearly this will be stopped short by famine, disease, nuclear war, or something. But the notion of "sustainable growth" of only a few percentage points per year is the crazy-talk of someone who doesn't realize the math. Radiation levels, population levels, CO2 omissions and many other things operate at large scales and are very real problems. They can't be addressed unless they can be understood in the first place, and that means it's important for the average person to have some grasp of exponential numbers.
Aaronsons's article also has a very fascinating section about WHY people can't grasp large numbers. Apparently, when you are asked to just guess "which of these two answers is probably closer" in a simple math problem, this uses a different part of the brain than when you have to compute something exactly. The estimation part of the brain is great for telling if there are about 3 or about 6 enemy tribespeople attacking you. But it's useless at the scale of exponential numbers. Humans use the language center of the brain to exactly compute these types of numbers, because no one has the intuition to deal with them.
I'll close with this quote from Aaronson:
"If people fear big numbers, is it any wonder that they fear science as well and turn for solace to the comforting smallness of mysticism?"