I think two things are (or were until people knew this stuff) reasonably surprising. I'll use weather as the example for both (since I work somewhere that does weather prediction).
Firstly, it is not intuitively obvious just how badly things blow up. I don't think it's easy to understand, unless you already know, that weather prediction two days out might be completely tractable with a computer you can more-or-less buy off the shelf, while weather prediction a month or a year out might require more computing resources than will ever exist in the universe. Indeed, I think that the existence of problems that have that kind of blow-up is one of the great discoveries of the 20th century: I don't think that, if you asked someone in 1900 whether such problems existed at all, they would have said yes: but it turns out that not only do they exist as fairly abstract computational problems, they are really common in quite mundane physics.
Secondly, and perhaps even more surprisingly is how simple these systems can be. Of course weather prediction is hard: there are a huge number of variables in the system so it's not surprising it's hard. But it is surprising that the most absurdly simplified model of weather it is possible to imagine -- a system with three variables evolving over time -- is also effectively unpredictable. If that is not surprising to you then you have lost your sense of wonder, because it is a deeply astonishing fact.