Benford's Law makes wonderful sense in situations of exponential growth.
Imagine you put $100.00 in a savings account that earns you 10% interest a year. At the end of the first year you have $110.00, at the end of the second you'll have $121.00, and the third will leave you with $133.10. The leading digit will remain a one until the eighth year (at which point you'll have $214.35). Two will be the leading digit for the next four years (at the end of which you'll have $313.84). Three more years will get you into the four hundreds (with $417.72), but you'll reach the five hundreds only two years after that. The more money you have in your account, the less time you'll spend with any particular leading digit. That is, until you've more then $1000.00 in your account. At this point it will again take you eight (or so) years to get to $2000.00, four to get to $3000.00, three to $4000.00, et cetera. A similar ratio from year to year will be present regardless of how high the interest rate is.
It is thus makes perfect sense that a "random" sampling of saving account balances will have about twice as many ones as their leading digits as twos, since the average account will spend almost twice as much time with a one as its leading digit. If we calculated the above for continuously compounded interest the numbers would match those predicted by Benford’s law even more closely.
Thus we expect things like the size of cities and the price of stocks to follow Benford’s Law, since both also grow exponentially. What’s freaky is how many unexpected things also follow the law; apparently logarithmic scales are more popular in nature and society then we might think.