I can totally see it being non-trivial to count your videos, which is a funny problem to have. But I doubt it's unknowable. More like they don't care/want us to know that.
But knowing the exact number can indeed be hard. It would take stopping the entire uploading and deletion activity. Of course they may have counters of uploads and deletions on every node which handles them, but the notion of 'the same instant' is tricky in distributed systems, so the exact number still remains elusive.
But the question of the precise number of videos before that moment is indeed ill-defined.
It's like knowing the distance to the Sun is 93 million miles. The difficulty there isn't that measuring the distance from the Sun to the Earth exactly is hard, although it is, or that the distance is constantly changing, although it is, or that the question is ill-defined, because the Earth is an object 8000 miles across and the Sun is 100 times bigger, and what points are you measuring between?
The distance is "unknowable" because while we know what "93 million miles" means, it's much harder to say we know what it "means". Even when we try to rephrase it to smaller numbers like "it's the distance you could walk in 90 human lifetimes" is still hard to really feel beyond "it's really really far."
Likewise, does it matter if YouTube has 100, 1000, or 10,000 millennia of video content? Does that number have any real meaning beyond back-of-the-envelope calculations of how much storage they're using? Or is "500 years per minute" the most comprehensible number they can give?