Grammar: using 的 several times
"He's the guy who just employed a person who doesn't even know how to work."

we translate to 他就是雇用了那個連工作都不知道如何做的人的人(傢伙)。


the girlfriend of the guy from the shop whose dad was a plumber.

we translate to 店裡那傢伙的女友的爸爸是水管工


How many 的 am I allowed to put in ?
You can put whatever you want but the sentence need to to reasonable.

What are the 的 I can remove ? ()means can omit

他(的)哥哥的新(的)手機的價錢是多少?


How come I never see this sort of thing in Chinese ?
This sort of thing is lengthy.we say 他哥新手機的價錢是多少? is same as previous answer.

Do you have any advice to cope with this sort of grammatical conundrum in the future ?
It is not a big problem for you to communicate with other because people like simple.