The crossed product is a fundamental construction of II$_1$ factors, and a natural question is whether every separable II$_1$ factor arises from a noncommutative dynamical system in this way. I will discuss joint work with Adriana Fernández Quero and Adrian Ioana in which we construct the first separable II$_1$ factors that cannot be decomposed as $B\rtimes G$ for any trace-preserving action of any countably infinite group $G$ on any tracial von Neumann algebra $B$. Our approach relies on a rigidity phenomenon for embeddings. Specifically, for the constructed factors, all embeddings into their tensor products come from the canonical embeddings, which is incompatible with the comultiplication associated to an infinite crossed product decomposition.