This is a great question. A basic calculus fact is that when you integrate, you get a continuous function. If you integrate a "nice function" (say, it's continuous) then you get a differentiable function. This makes sense, since if you differentiate, you get the function back. In this sense, you are basically "adding" one derivative to the function with each time you integrate.
Conversely, if the original function were not differentiable, then if you started differentiating, you would get one step closer to the non-differentiable original function with each differentiation. So in that sense, the function is getting worse, because it is "losing" derivatives. This will often add more and more "bad" behavior like rapid wobbling, because without more derivatives (which have finite values everywhere they are defined), there is less control on the wobbling.
Functions that you can keep differentiating forever and still have more derivatives (however "bad"), are called "smooth". These are incredibly rare and special functions. Their rareness is not apparent in basic calculus classes because x and e^x are the only "words" we use to write almost all functions we care about, and they are both smooth. (Is it surprising a course about derivatives deals in functions with derivatives?)
There is an entire field of math called "regularity", which basically means the niceness of a function. Often that boils down to how many derivatives a function has, and whether they are bounded. (This is important for finding solutions to differential equations, and applied fields like numerical analysis. Speed of convergence for a numerical algorithm generally depends the size of some derivative). Just a warning, it's a deep rabbit hole. There is more nuance to even what I just said, with varying levels of differentiability (there are half derivatives and weak derivatives), different levels of continuous (eg continuous < uniformly continuous < absolutely continuous < differentiable), and stranger notions like "almost continuous" and "differentiable almost everywhere". It's very interesting and applicable, but very hairy.