Last night an old idea came back to me, an idea about a function where all the derivatives start from zero and then grow smoothly. I thought it would be impossible, but then I found some interesting stuff on Wikipedia. So, I learned to use SymPy and wasted a lot of time with it. Here's a report of my (non-)findings.
(UPDATE: I did some numerical differentiation, which showed that h(x) does have negative derivatives. See details in this comment. A disappointment, although perhaps not a surprising one. It doesn't however, necessarily mean the goal is impossible.)
So, if anyone knows whether such a function exists and what it looks like, please tell me.
I did some numerical differentiation, with ten thousand points between 0 and 10. Negative values appeared in the third derivative. The attached figure zooms into them. While I think those sudden spikes may very well be numerical artifacts caused by float rounding errors or something like that, there is a clear negative slope around them, further confirmed in the fourth derivative. So, this function is not what I hoped it to be.