During the experiment, the scientist continued on, despite the risks. Which words in the sentence are objects of the underlined preposition?

Despite the risks is a prepositional phrase. despite is the preposition. The noun (in this case) that is an object to a preposition is risks. As far as I can see, that's the only prepositional phrase. Many times "on" is, as in "The book is on the table", but not in this case. On in this case is an adverb. It has the same usage as later on, or carrying on.