Can the third derivative tell me anything about the original function? 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?
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).
When in college, I was told that the order of the derivative is related to the "dimension." 

I think this is easier to see once you take integration. Double integrals give us regions in the two dimensions while triple integrals give us regions in three dimensions.