Despite being correct in the narrow sense, you nevertheless end up being wrong in a broader, more useful sense. I do not disagree with the definition. But you

Despite being correct in the narrow sense, you nevertheless end up being wrong in a broader, more useful sense. I do not disagree with the definition. But you shouldn't rigidly focus on the definition without considering the utility of the notation, and the purpose of the discussion. Big O notation describes the asymptotic behavior of a function when the argument N being passed to it approaches infinity. This is why it is also known as asymptotic notation. Now, since this thread is about practical consequence, there is a severe problem with glibly saying that n * 999^999^999^999^999 will ever remotely resemble n from a practical standpoint. You'll never be able to compute f(1), much less a value of n large enough to be limiting. At any rate, it looks like we both agree that O notation doesn't help you here. My fault for bringing it up.