[schoen]

Heh, that gives this nice expression

  sum(n>=10**i for i in range(100))

for (for example) inputs up to a googol. Or with your fix for the base case,

  1 + sum(n>=10**i for i in range(1, 100))

Another cute way that hides the loop

  from itertools import count, takewhile
  1 + len(list(takewhile(lambda x: 10**x <= n, count(1))))

[kristopolous]

a way that follows the rules and also does what you're trying to do:

    def noloops(n):

      x = lambda n, digits: (n / 1e9, digits + (n >= 10) + (n >= 100) + (n >= 1e3) + (n >= 1e4) + (n >= 1e5) + (n >= 1e6) + (n >= 1e7) + (n >= 1e8) + (n >= 1e9))

      y = lambda n, digits: x(*x(*x(*x(*x(*x(n, digits))))))

      return y(*y(*y(*y(*y(*y(n, 1))))))[1]

Now we have it up to 324 digits without any loops.

[psychoslave]

How is that hiding the loop? I would expect takewhile to be a loop although I never used this Python facility.

[schoen]

All of the itertools functions are implemented using loops, but they heavily abstract over them so that users can think in terms of "streams" (or officially "iterators") without writing loop-oriented code themselves.