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.
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.
[1] http://en.wikipedia.org/wiki/Busy_beaver