"There are not many restaurants around the hotel."
I am Chinese. “的” is to point out the possession, just like ('s) in English,
e.g. Wang's book - 王的書, Li's telephone - 李的電話。
Make a one to one translation:
酒店 - 的- 週圍 = hotel - 's - around
Of course it's not a structure of English, therefore we say "around the hotel".
For the example you mentioned "我的家周围有很多树". Imagine an example in English: Your mother has a private doctor. And the doctor has a daughter. Then when you mention the daughter, you won’t say “My mother’s private doctor’s daughter”. You will say “my mother private doctor’s daughter” (it may not be a proper English structure, but I just want to show you how the above Chinese sentence similar to it.)
Therefore, 我的家周围有很多树 or 我的家的周围有很多树 are not wrong. But 我的家的周围有很多树 is not so native. Just like “My mother’s private doctor’s daughter”.
Hope it can help you.
Thanks for the reply. So if I were to formulate a general rule, would it be correct to say that only the first 的 is kept in case of nested possessives? E.g. if I extend your example sentence a bit and nest three posessives: "My mother’s doctor’s daughter's house", would the most natural Chinese translation omit 的 twice like 我妈妈的医生女儿家 ?
The example you made is already out of my knowledge because we won’t say a sentence with such many nested possessives. And also, there is no general rule on whether the first "的" should be kept or the second one.
I can only say in the example you mentioned. “我的家周围有很多树” and “我家的周围有很多树” have the same meaning and native. But “我的家的周围有很多树” is not so native. It's not wrong, but it’s a little bit cumbersome. We won't say Chinese in that way.