I always wondered how fast the denominators of the harmonic numbers grow.
This is answered in: http://mathworld.wolfram.com/HarmonicNumber.html
It says that for n = 0, 1, 2, 3, ..., the number of digits in the denominator of HarmonicNumber[10^n] is given by 1, 4, 40, 433, 4345, 43450, 434110, 4342302, 43428678, ... (Sloane's A114468). These digits converge to what appears to be the decimal digits of log_(10)e==0.43429448...
I don't know about anyone else, but I think this is neat! (Is it obvious?)
Well, let's see. Call the denominator D(n). The number of digits is basically log10 D(n), and the claim is that log10 D(10^n) ~ 10^n log10 e, which (dividing through by log10 e) is the same as log D(10^n) ~ 10^n or log D(N) ~ N. Now, naively we'd expect D(N) to be something like gcd(1,2,...,N), whose exponent of p is approximately log N / log p so that log D(N) ~ sum over p of log N ~ (N/log N) log N ~ N. Of course I'm using "~" in a somewhat looser sense than is traditional and I haven't proved any of the potentially tricky bits. :-) -- g