The first derivative can tell me about the intervals of increase/decrease for $f(x)$. The second derivative can tell me about the concavity of $f(x)$.
So can the third derivatives, and any derivatives beyond, yield any useful piece of information for graphing the original function?
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It all depends on the function itself, because a linear function for example isn't concave in the first place. If you had a vector field represented by vector function "e.g. wind velocity", then you want to specify, are you talking about gradients or divergence or curl etc, as these are all derivation operators.
It might not be as useful as simply the second derivative.
But let's say you have a scalar field representing the elevation of a terrain, and you want to know how rough a terrain represented by 3d function, you can get an idea how rough it is from the third derivative. In a rough terrain, you will have a non-monotonic third derivative. Note, that this interpretation is strictly a creative way to look at it, as i don't have any reference supporting the previous.
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Yes: it tells you about the rate of change of the curvature of a plane curve, which is given by the formula $$ \kappa = \frac{y''}{(1+y'^2)^{3/2}} $$ The derivative of this is $$ \kappa' = \frac{y'''}{(1+y'^2)^{3/2}} - \frac{3y' y''^2}{(1+y'^2)^{5/2}}. $$
If you work in more than two dimensions, the torsion of a curve involves the third derivative: this tells you how non-planar it is (the helix has non-zero torsion, for example).
Through supremely literal, I guess just as we can think of the first derivative as 'how quickly the function changes' and the second as 'how quickly the function of how quickly the function changes changes', we can say that the third is just 'how quickly the function of how quickly the function of how quickly the function changes changes changes'.
I would say that concavity and slope only seem significant because we gave them very visual names. We gave them names because they were used in visualising functions, but they are nothing more than a geometrical interpretation of the above quoted statements.
If we insert the names of the previous iterations into those statements, we get that the second derivative is 'how quickly the slope changes' and the third derivative is 'how quickly the function of how quickly the slope changes changes', which is just 'how quickly the concavity changes'.
Maybe we can give this one a nice geometrical name too, something like 'flexion', because it's describing whether the concavity is becoming tighter or looser and how quickly it is happening.
But there's nothing particularly remarkable about this property, just as there is nothing inherently remarkable about slope or curvature (just that we have given them names).