We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by their inert maps. We show that left Kan extension along the inclusion $j \colon \Delta_{\operatorname{inert}} \to \Delta$ takes general objects to M\"obius decomposition spaces and general maps to CULF maps. We establish an equivalence of $\infty$-categories $\mathbf{PrSh}(\Delta_{\operatorname{inert}}) \simeq \mathbf{Decomp}_{/B\mathbb{N}}$. Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions. We show that the Aguiar--Bergeron--Sottile map to the decomposition space of quasi-symmetric functions, from any M\"obius decomposition space $X$, factors through the free decomposition space of nondegenerate simplices of $X$, and offer a conceptual explanation of the zeta function featured in the universal property of $\operatorname{QSym}$.