The degree operator too optionally combines with an infinitival clause, and this infinitival clause can contain a non-subject gap: John is too rich [for the monastery to hire (him)]. If the object pronoun in this example is read as anaphoric to John, the sentence has the same meaning with or without the gap. This paper begins by asking both what makes it possible for infinitivals with too to contain non-subject gaps, and how the same meaning arises with or without the gap. In his (1977) paper On Wh-Movement, Chomksy argued for an approach to the first question in which the non-subject gap in an infinitival with too is a trace left behind by a null operator that has moved to the edge of its clause to form a predicate of individuals. While Chomsky did not address the second question, he did assume that the infinitival is a complement of too. Under this assumption, a straightforward extension of Chomsky's proposal would posit two homophonous degree operators too: one that takes a propositional infinitival clause as its complement and one that takes a property-denoting infinitival. This paper spells out such an analysis, but then presents two types of surprising observations that it fails to predict, both having to do with the interpretation of too when it appears under intensional verbs. The first of the puzzles appears to be a restriction on the logical scope of the degree operator, while the second appears to be a restriction on the binding of world variables. The two puzzles are left as an empirical challenge that must be met by a successful theory about gapless infinitivals with too.