Theoretically, the sentence isn't incorrect. An English speaker could understand it. However, it doesn't sound very well-structured and it would not be a stellar example for non-speakers.
Here are a few changes you might like in order to make it sound better:
but it is not as dramatised as Cook's case
Simply removing "as much" makes the sentence sound less wordy and more pleasant, and it retains the meaning of "it is not dramatised to the extent that Cook's case is."
Another possible change could be:
but it is not dramatised as much as Cook's case
The first "as" does not need to be there to keep that same intent for the sentence.
If you do that, you may want to (though it is not a legal necessity, it clarifies, and might even sound better) append an "is" to the very end of the sentence, like:
but it is not dramatised as much as Cook's case is
Now, here's why:
Your original sentence was as follows:
but it is not as dramatised as much as Cook's case.
Now, the first "as" and the "as much" are somewhat interchangeable. By that I mean that if you use "as _____ as _____," then there is no necessity for the "as much" before the second "as." Or, if you would rather use "as much," then the first "as" is not necessary: "_____ as much as _____ is." The final "as" needs to stay where it is because it indicates the second object of comparison. Using "as ____ as ____" is usually for a comparison, for example:
The woman is not as tall as the man.
Using "____ as much as ____" is often for a contrast including a verb. For example:
He is not loved as much as she.
They cost as much as these.
With that in mind, either of these can work. But, to answer your original question, no. You should not use all three in a row.