Can Fregeans have 'I'-thoughts?

We examine how Frege's contrast between identity judgments of the forms " a=a " vs. " a=b " would fare in the special case where 'a' and 'b' are complex mental representations, and 'a' stands for an introspected 'I'-thought. We first argue that the Fregean treatment of I-thoughts entails that they are what we call " one-shot thoughts " : they can only be thought once. This has the surprising consequence that no instance of the " a=a " form of judgment in this specific case comes out true, let alone a priori true. This further reinforces Glezakos's objections against the setup of Frege's puzzle, while also raising what we think is an acute problem for Fregeans, insofar as I-thought (and indexical thinking more generally), understood in their way, turns out to be incompatible with some basic features of rationality.