Can the third derivative tell me anything about the original function?

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).

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).