Sorry about my misreading your example, Nino. It looked so bizarre that unconsciously I must have interpreted it as a mistake. (Maybe a useful sign, though!

) Thank you, for your links, I haven't heard about this phenomenon before.
However... in your wiki link, there is something to consider specially:
"For morphological causatives, some language do not allow single morpheme to be applied twice on a single verb (...) while others do (...
Hungarian...), though sometimes with an idiomatic meaning."
Apart from the fact, that -at/-et and -tat/-tet are
different suffixes (albeit both for the causative), I think (and
that would be interesting to look into) that in Hungarian it is likely that the usage is closer to an idiomatic one because I think we can all agree that even if there are perfectly valid examples for verbs having these two causative suffixes (főzettet, írattat stb.), native speakers do not recognize any double causative meaning behind them. (At least at first sight and without a context.) Or even if they do, expressing a third person doesn't seem to be very important. If it is, you wouldn't express it by a double causative but by introducing another structure.
And just an example to illustrate this:
Your sentence: "A tanár dolgozatot írattat a helyettesítővel." is ambiguous, although
mainly in this present context.
If I just heard it, I would think that it is the
helyettesítő* who has to write a test but even if it wasn't the case, I probably wouldn't look any further to find out who did what to whom exactly.
The other alternative: the teacher asked the replacement to give a test to the pupils.
But if you really meant that, you'd probably wouldn't formulate your sentence like that.
It would be more natural (and clearer) to say e.g.: A tanár megkérte a helyettesítőjét, hogy írasson dolgozatot (a gyerekekkel).
I think the first variant (A tanár dolgozatot írattat a gyerekekkel) is somewhat different (to start with: sounds more natural) but then the double causative is totally "useless" if it is him who gives the test to the students...**
*You'll find the special characters among the tools, clicking on the omega sign (last in the row)
** Maybe a little bit like in English in a sentence like: The pupils were given a test - by whom is not important. (Because it is either obvious, the teacher, or irrelevant in the moment of speaking.)