Do nested complements exist in Chinese?
Short answer: yes.

I think we should at least distinguish two types of complements.

For the first type, it is very difficult to insert anything else between the complement and the verb. One way to think of them is to view them as the inflection of verbs.

✔说得出话 verb + de + complement + object
✔也说得出话 adverb + verb + de + complement + object
✘说也得出话 verb + adverb + de + complement + object
✘说得也出话 verb + de + adverb + complement + object
✘说得出也话 verb + de + complement + adverb + object
✔说得出话也(说不清楚)verb + de + complement + object + adverb


On the other hand, the complement introduced by 得 has more freedom. We can make up the following grammatical, though by no means usual, sentence:

✔他 长得 丑得 看得我 吓得 紧张得 说不出话 来


When I interpret this sentence, I tend to think of 得 as a kind of postposition, or a conjunction. For example, I parse the long sentence as:

他长得很丑。丑得看起来吓人。我吓得很紧张。紧张得说不出话来。


In this way, you can flatten the sentence so that you will not need much short-term memory to be able to understand it. I do not consider the second 得 as inner, rather, I think it is outer, or at least parallel, because the part following 得 sounds more like the main clause. For example, interrogatives and exclamations are allowed after 得.

✔他 长得 好不好看? interrogative 
✔他 长得 真好看! exclamation


It can be demonstrated that adverbs that are normally attached to subjects can be inserted after 得:

✔她 长得也 很漂亮，穿得也 很漂亮


In deed, the above verb + 得 structure resembles nouns and verb phrases:

✔她 读书 很好 verb + adjective
✘她的读书 很好 dushu is not a noun
✔她 长得 很漂亮 verb + de + adjective
✔她(的)长相 很漂亮 noun + adjective


And I do not find it particular unnatural to coordinate these phrases of different classes:

✔她 长得 和 穿着 都 很好 [verb+de] and [noun]


I think this somehow explains the reason even native speakers do not feel much difference between the following sentences:

✔他 说得 是什么
✔他 说的 是什么
✔他 说的 什么
✔他 (是)怎么 说的