Is it possible to set a limited amount of extra ingredients, so that customers cannot choose more than 5 ingredients

assuming you are talking about maximum 5 of (let’s say) 10 available, then yes .
obviously , if you are talking about you only want 5 available to choose from, then you just offer those 5 ingredients)

drop-down menu…

no, as you will run into serious usability issues. However:

a) you can edit/amend the css to have – for example – 4 ingredients per row as opposed to the default 3 (whether that looks any good depends on your theme though)

b) you can easily write some javascript that only ever shows parts of the ingredients or set the height of the div that contains the ingredients and make it scrollable or whatever else you can think of