Symmetric tensor categories in characteristic 2

We construct and study a nested sequence of finite symmetric tensor categories ${\rm Vec}=\mathcal{C}_0\subset \mathcal{C}_1\subset\cdots\subset \mathcal{C}_n\subset\cdots$ over a field of characteristic $2$ such that $\mathcal{C}_{2n}$ are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius--Perron dimension. This generalizes the category $\mathcal{C}_1$ described by Venkatesh and the category $\mathcal{C}_2$ defined by Ostrik. The Grothendieck rings of the categories $\mathcal{C}_{2n}$ and $\mathcal{C}_{2n+1}$ are both isomorphic to the ring of real cyclotomic integers defined by a primitive $2^{n+2}$-th root of unity, $\mathcal{O}_n=\mathbb Z[2\cos(\pi/2^{n+1})]$.