for 他一叠声应着 I know what yidiesheng means repeat the sound, but I just wanted to know the structure is die a measure word here? A pile of sounds? It doesn't actually say anything like that in the dictionary, just the 2 definitions being pile up and repeat and fold. So what does the 一 do here?
叠 here is acting as a classifier for 声
Normally, 叠(stack of) is a classifier for things that you can stack up on top of each other, like a stack of paper or a stack of boxes.
声 (sound) of course, is not physically stackable. However, stacking also means putting things on top of each other ' one after another (continuously)' and things that are stacked are 'tightly packed (no pause)'
It is a metaphorical usage of the classifier 'a stack of' to describe a series of vocal sounds in quick succession
他一叠声应着 = 他一叠声(地)应着
The subject is 他 = He
The adverb is 一叠声(地) = continuously/ repeatedly
The verb is 应着 = responding
By 《漢語大詞典》, 疊聲 means '一聲緊接一聲', which can also be written as '迭聲'. From the quoted examples, 迭声 seems older.
《水浒传》第九八回：“兩軍相撞，擂鼓摇旗，兩陣裏迭聲呐喊。” (Ming Dynasty)
疊聲 and 迭聲 sound very similar (exactly the same in Putonghua) and means the same. (Note that 迭 means 轮换、交替, 叠 means 累积、折叠, they usually cannot be used interchangeably《現代漢語規範詞典》. But in this word, both are used) Since 迭聲 is attested in much older texts, it's highly likely to be the original version.
By 《現代漢語規範詞典》, 迭 has the verbal usages of '轮换；交替' and '赶上；赶得上（常用于否定）', and the adverbial usage of '屡次'. By 《王力古漢語字典》, the primary meaning of 迭 is 更迭，輪流 (verb), from which 接連，屢次 (verb used adverbially > lexicalized as an adverb) and 連上，够得上 (verb, extension of meaning) are derived. It's also used as 叠, means 堆積，重叠 (verb). Examples given are from Song and Qing Dynasties. We can conclude that although in modern Chinese, 叠 and 迭 cannot be used interchangeable, in the eras when 疊聲/迭聲 were created, 迭 can be used as 叠, and this set of words are left as a legacy in modern Chinese. 迭/叠 are used as a verb in this word. (Adverbs cannot directly modify nouns in Chinese) Either interpreting it as 堆積，重叠 or 更迭，輪流 is both fine. This view is further corroborated by 冰心's example, where it is used as a predicate, interpreted as a VP (verb+object).
'一連疊聲' should already exist before 冰心's usage, but could it the source of 一疊聲? I don't know, but think it's not likely.
一疊/迭聲 loses the verbal usage of 疊/迭聲 and can only be used adverbially. Compared with the adverbial usage of 疊/迭聲, it's stronger and exclusive. Ex., 《紅樓夢》第六五回：「尤三姐一疊聲又叫：『將姐姐請來，要樂，咱們四個一處同樂。』'Stronger', it's used as a particle to intensify the mood. By 'exclusive' I mean 一 excludes other manners that 叫 is performed. In this sense, 一 is used adverbially, meaning 全，都. Since all these usages are derived from its primary numeral usage, and just presents different levels of 虚化(grammaticalization, the intensifying particle usage is more grammaticallized), the usage in this word is very likely mixed.
Synchronically analyzing 一疊聲 as numeral+classifier+noun is both grammatically and semantically correct. But from the diachronic analysis above, as well as the existence of 一迭聲 and the fact that 迭 is not used as a classifier in modern Chinese, I vote for the verbal usage.
Interestingly 一疊/一迭 are used in literary Chinese to mean 1)古代以吹角十二聲為疊 (quoted examples from Tang and Ming Dynasty), 2)猶一曲 (quoted example from Song Dynasty) (《漢語大詞典》). There, they were still nouns, not grammaticalized to classifiers yet.
UPDATE: RESPONSE TO @PdotWang
All the three editions of《漢語大詞典》treat 一迭聲 as another writing of 一疊聲. This strongly goes against the classifier view.
Though not the 尤三姐 quote, the 他忽然性急起來，一疊聲叫‘快！快一點！’ quote is essentially the same. It's explicitly written in this entry that 一疊聲 is also written as 一迭聲. This set is not distinguished.
It does distinguish 疊聲/迭聲: 疊聲 can be written as 迭聲, but some of the usages of 迭聲 are not written as 疊聲 (due to the semantic difference of the verbs 迭 and 疊, I think).
@PdotWang points out the 叠 had once been 迭's simplified character. Um, it complicates the discussion a bit. But 《漢語大詞典》 is written in traditional Chinese, so it shouldn't be a problem. Also, I checked 《现代汉语大词典》 , which is written in simplified Chinese but well after 1986. Interestingly it only includes 一迭声 but not 一叠声.
他一叠声应着。 = 他连连答应着。= He said a few times that he would do it.
Yes， 叠 is a measure word (a noun) here, just like in 一叠纸, 一叠卡片，一叠盘子. It usually indicates the piled-up of the same things.
For example, 公路的尽头是重重叠叠的高山峻岭。= There are mountains and mountains at the other end of the highway.
叠 is also a verb, = to fold over, such as 她用纸叠了一个小船。= She made an origami boat. = She folded a piece of paper into a small boat.
There is another verb 迭, which is a homophone word of 叠, which means to revolve (to come out in turn).
For example, 四季迭换。= The four seasons go around one after another.
不迭 means no-stopping, such as 叫苦不迭。= Could not even wait for the former one to end. = Complaining endlessly. = One complaining after another.
Another homophone word is 喋 as in 喋喋不休。= talkatively.
For example, 他喋喋不休地讲着同样的故事。= He was telling the same old story again and again.