  We discuss the classification of strongly regular vertex operator algebras
(VOAs) with exactly three simple modules whose character vector satisfies a
monic modular linear differential equation with irreducible monodromy. Our Main
Theorem provides a classification of all such VOAs in the form of one infinite
family of affine VOAs, one individual affine algebra and two Virasoro algebras,
together with a family of eleven exceptional character vectors and associated
data that we call the $U$-series. We prove that there are at least $15$ VOAs in
the $U$-series occurring as commutants in a Schellekens list holomorphic VOA.
These include the affine algebra $E_{8,2}$ and H\"ohn's Baby Monster VOA
$\mathbf{VB}^\natural_{(0)}$ but the other $13$ seem to be new. The idea in the
proof of our Main Theorem is to exploit properties of a family of vector-valued
modular forms with rational functions as Fourier coefficients, which solves a
family of modular linear differential equations in terms of generalized
hypergeometric series.
