  The construction of Quasi-topological gravities in three-dimensions requires
coupling a scalar field to the metric. As shown in arXiv:2104.10172, the
resulting "Electromagnetic" Quasi-topological (EQT) theories admit charged
black hole solutions characterized by a single-function for the metric,
$-g_{tt}=g^{-1}_{rr}\equiv f(r)$, and a simple azimuthal form for the scalar.
Such black holes, whose metric can be determined fully analytically, generalize
the BTZ solution in various ways, including singularity-free black holes
without any fine-tuning of couplings or parameters. In this paper we extend the
family of EQT theories to general curvature orders. We show that, beyond linear
order, $f(r)$ satisfies a second-order differential equation rather than an
algebraic one, making the corresponding theories belong to the Electromagnetic
Generalized Quasi-topological (EGQT) class. We prove that at each curvature
order, the most general EGQT density is given by a single term which
contributes nontrivially to the equation of $f(r)$ plus densities which do not
contribute at all to such equation. The proof relies on the counting of the
exact number of independent order-$n$ densities of the form
$\mathcal{L}(R_{ab},\partial_a \phi)$, which we carry out. We study some
general aspects of the new families of EGQT black-hole solutions, including
their thermodynamic properties and the fulfillment of the first law, and
explicitly construct a few of them numerically.
