The problem is that English grammar, as it has always been taught, is based on Latin grammar. And Latin and English are very different languages.
I'd say this is not a simple sentence. In Latin terms, "hoping to win $1000, $100, or even just a trophy" is a subordinate clause, an adverbial clause.
It's subordinate because it depends on the main clause.
It's adverbial because it answers the question "why" they test.
I don't know how traditional grammar would analyze "hoping to win", and I don't really care. To me this is common sense:
The English infinitive is really more like a noun than a verb.
"to win" is the object of hoping, so it's behaving like a noun.
But "to win" as a verb takes the object "$1000..."
As you know, it's often hard to slot English words into categories like noun, verb, adjective, etc.
This isn't meant to be the final answer to your question; I hope others will respond with their opinions.