Handsomest is the superlative form of the adjective "handsome".
There are three degrees of adjectives- positive degree, comparative degree and superlative degree. For the right degree to be used along with the sentence, the superlative degree of the word 'handsome' will be the best used n this sentence. And according to the sentence, Rob is in comparison with the 'rest of the world, making the comparison to be the third degree of comparison. Thus, the final answer will be
Rob thinks that he is the <u><em>handsomest</em></u> guy on the planet.
