A related anecdote: I saw mathematician Andrew Dilnot [1] at a Mathematica users conference a couple years ago, and he asked the audience how many times the size of the world economy had doubled since 1900, given that it had grown at about 5% per year. The audience guesses were in the 2-4 range, but he pointed out that the rule of 72 gives 110/(72/5) ~= 7.6. "Don't feel bad," he said, "last week I gave this talk to the council of European finance ministers, and they did no better!" (I may not have remembered the numbers exactly, but they were something like that.)
Your calculation is way off. You forget that the rule of 72 is how long it takes for the money to double. Thus, the final calculation is 2^7.6 which is, as any good computer scientist knows, a little less than 256
Huh? At 5% growth per annum, the economy takes 70/5=14 years to double (I prefer the rule of 70). 112 years since 1900, so that's enough time for the economy to double 112/14=8 times.
The economy grew by a factor of 1.05^112 = 236. Which is close enough to his number to assume that he missed the "doubling" part of the question. 236 is about 7-8 doublings.
Woops, yeah, I thought it asked how many times larger the economy was, not how many times it doubled (which, btw, seems like a weird way to phrase the question to me).
[1] http://en.wikipedia.org/wiki/Andrew_Dilnot