1000^depth is not a constant: the function here is "f(depth) = number of times the inner loop body executes". f(depth) = O(1000^depth).
And, "times" here is serving the same role as "comparisons" in "Mergesort takes O(n log n) comparisons": it's referring to the thing that f is actually counting.
Yes, O(…) is just a mathematical construct, so you can apply it the way you have to mean "the amount of work done for a given input size increases exponentially with respect to the number of nested loops iterating over the input"… but that's not how most people interpret it in the context of a computer algorithm. (The exception would be if the nesting depth were dynamically determined by an input parameter, but I don't think that's what anyone here is talking about.)
Anyway I'm not trying to argue, I agree with your point, but I think everyone here is talking past each other trying to say the same thing, ultimately due to imprecision in semantics.
And usually when you encounter O(some constant), it's meant as "of the order of magnitude of", i.e. somewhere between some constant/10 and some constant * 10. That isn't the definition used here, but seems to be the cause of most complaints about asymptotic analysis being misapplied when no asymptotic analysis was being done in the first place.
