At what point does it make sense to add a search box to a dropdown list, and does the styling of the list affect that decision?
For example: Is 15 elements is a good number when each element has a height of 50px?
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I don't think there is a specific answer for the question since it will definitely depend on the context and various requirements or constraints.
However, you can reference examples like the searchable dropdown lists used for tasks like searching for a country or looking up an address.
If the search algorithm and autocomplete feature is smart or powerful enough to handle the query, then it is certainly useful when the number of items in the list is not easily scannable or can't be narrowed down easily.
I would suggest understand the search and browse behaviour for your application and the trade-offs involved before working out when and how to implement this feature.