As stated in my last entry, I am now able to compute billions of continued fraction terms for arbitrary constants. In fact, using my current setup, I have already charted over three billion such terms for the constant π. (Because of memory constraints, an attempt at four billion terms failed.)
There are a number of things an empiricist can do with such a collection: tally it (these occurrence counts are for exactly 3 billion terms), find the position of first occurrence, and the position of the nth occurrence, of n. (All three of these exclude the initial 3, because the initial — the zeroth — term of of a simple continued fraction may be any integer but the rest are strictly positive. This is why I do not like the Applications example in the Khinchin entry of the Mathematica documentation center.)
Additionally, what I did back in 2001 (with a measly 53 million terms) was generate a π Khinchin-approach sequence: A059101, with 27 terms. Today I added terms #28 to #36.