It seems like the real character here is Graham, who I've never met, but I do have a story about his collaborator Bruce Rothschild, who was my professor for junior-level linear algebra.
In those days, you either gave your professors little self-addressed stamped postcards to tell you your grade, or went to their offices to ask them. I took the latter approach with Rothschild. "What grade were you expecting?", he replied. I confessed that I thought I would probably get an A. "You're getting an F", he said. "I'm giving everybody the opposite of what they expect."
I TA'd for Graham's Discrete math class at UCSD. He's still quite the character! His last lecture he juggled a handful of chalks in class and brought his deck of cards to do magic tricks.
He once made me pull out a 1$ bill from my pocket in his office and then asked me what the serial number was, he shot out a question like: "what's the probability that you have n consecutive numbers on the bill".
You could tell Math was legitimately fun for him, he treated open math problems like brain teasers and that fun rubbed off on the people around him as well.
No, that's something to do in a Hilary Putnam philosophy class. He's the one who announces at the beginning of the term there will be a two-part final:
A. Write a question suitable for this course.
B. Answer it.
You will be graded on both parts.
He wasn't serious but he would have been interested in a quick answer. I didn't get it cause I couldn't think of a quick way to know what a reasonable upper bound for the serial number is (I believe they're printed sequentially). To be fair the guy spent his whole career learning how to count things!
There was a contest to write the 512-char C program that generated the largest number (but still terminates), assuming that all integer types are unbounded: http://djm.cc/bignum-results.txt
The method used by the winning entry is extremely interesting: return the outputs of all possible programs up to a certain size which terminate according to the calculus of constructions. It's quite crazy to have done this in 512 chars.
Graham is the coolest UCSD professor I have ever had a conversation with. I remember a particular instance, at the beginning of my first quarter at UCSD. I wasn't familiar with his work and had never heard of him. My naive 19 year old self walked into his office hours asking some simple probability question. I noticed a cast on his hand and inquired about it. He jokingly said he got a cut because he was juggling ice skates. I thought he was joking. I went home and looked him up, only to find that he was the head of the national jugglers association at one point. Pretty awesome guy.
I love this video http://www.youtube.com/watch?v=1N6cOC2P8fQ by Sean Day[9] Plott describing Graham's number very well. At least I found that description very easy to follow. Numberphile [1] too has a great explanation, but I still strongly recommend watching Day[9] video because it tells a little bit of history and Sean is very expressive and good speaker so he easily engages viewer.
I think fast growing functions could be interesting as well. My favorite example is the "busy beaver function"[1]. It has been shown that it grows faster asymptotically than any computable function, where computability roughly means that you can write an algorithm to calculate its value (without memory or running time constrains).
Note that the Graham number can be written as G=f[64](4), where [.] means the number of iterations of f and f(4)=3\up\up\up\up3 (as in the article). Now you can define a function g(n) as g(n):=f[n](4), so G=g(64). g is computable since we just described an algorithm to calculates its values. So the busy beaver grows even faster (asymptotically) than g.
"[T]he observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume."
In those days, you either gave your professors little self-addressed stamped postcards to tell you your grade, or went to their offices to ask them. I took the latter approach with Rothschild. "What grade were you expecting?", he replied. I confessed that I thought I would probably get an A. "You're getting an F", he said. "I'm giving everybody the opposite of what they expect."