  We consider a class of manifolds $\mathcal{M}$ obtained by taking the
connected sum of a finite number of $N$-dimensional Riemannian manifolds of the
form $(\mathbb{R}^{n_i}, \delta) \times (\mathcal{M}_i, g)$, where
$\mathcal{M}_i$ is a compact manifold, with the product metric. The case of
greatest interest is when the Euclidean dimensions $n_i$ are not all equal.
This means that the ends have different `asymptotic dimension', and implies
that the Riemannian manifold $\mathcal{M}$ is not a doubling space. We
completely describe the range of exponents $p$ for which the Riesz transform on
$\mathcal{M}$ is a bounded operator on $L^p(\mathcal{M})$. Namely, under the
assumption that each $n_i$ is at least $3$, we show that Riesz transform is of
weak type $(1,1)$, is continuous on $L^p$ for all $p \in (1, \min_i n_i)$, and
is unbounded on $L^p$ otherwise. This generalizes results of the first-named
author with Carron and Coulhon devoted to the doubling case of the connected
sum of several copies of Euclidean space $\mathbb{R}^{N}$, and of Carron
concerning the Riesz transform on connected sums.
