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I've been learning about these for awhile, but everything I've seen them used in doesn't use more than 1 at a time. Is there a reason for that?
Also my materials don't really explain the subtleties behind using 込む(to do something in an upfront way, i.e.聞き込む) and 回る(to go around doing something, i.e. 歩き回る) as 助動詞. Is there anything more I should know?
As @snailboat and @Chocolate have noted, I think you are actually thinking about something called 複合動詞{ふくごうどうし} (or compound verbs.)
It is interesting to consider combining more than two verbs when creating 複合動詞{ふくごうどうし}... and it certainly is possible, according to the example word lists given in this paper by 林 翠芳 (LIN Cuifang). LIN gives examples of times where 複合動詞{ふくごうどうし} can involve the process of combining three separate verbs into one compound verb (三次結合複合動詞{さんじけつごうふくごうどうし} vs 二次結合複合動詞{にじけつごうふくごうどうし}).
For example:
書{か}く＋立{た}てる＋すぎる ＝ 書{か}き＋立{た}て＋すぎる ＝ 書{か}き立{た}てすぎる
立{た}つ＋止{と}まる＋かける ＝ 立{た}ち＋止{ど}まり＋かける ＝ 立{た}ち止{ど}まりかける
So in answer to the first question, while there are times where three-verb-combined verbs exist, it's possible that the reason two-verb-combined verbs are more commonly seen is because the words have a simpler meaning or definition. In other words, the more convoluted a word becomes, the more complex its meaning.
In answer to the second question, there has been at least one similar question asked in regard to the meaning of ー込{こ}む when used at the end of a verb. Though the meaning may at times go beyond what is listed in the current list of answers for that other question, I would still recommend reading that other question in order to get a better idea of any additional subtleties related to ー込{こ}む.
Finally, verbs ending with ー回{まわ}る are fairly straightforward (typically being related to something "around" or "about", as you noted.)
As far as 複合動詞{ふくごうどうし} go in general, I would recommend taking a look at this paper when time permits. The paper better explains different functions, meanings, and aspects of 複合動詞{ふくごうどうし}.
