Does the Weeks manifold have the smallest volume among all finite volume oriented hyperbolic 3-manifolds? It is a result of Chinburg-Friedman-Jones-Reid that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold.
There is also a result of Milley that says that if $N$ is a closed orientable hyperbolic 3-manifold with volume less than or equal to that of the Weeks manifold, then $N$ is homeomorphic to the Weeks manifold.
My question is, therefore, is it possible to have a non-arithmetic, non-closed, orientable hyperbolic 3-manifold with volume smaller than that of the Weeks manifold? I am also curious if increasing the number of cusps necessarily increases the volume, as this would be sufficient to answer my question.
 A: The answer given in the comment by Ryan Budney should be enough, but let me give you a couple of theorems if it is still not so clear.
Suppose you have a finite volume hyperbolic complete orientable manifold $M$. Suppose that $M$ has cusps (hence, it is not compact).
Theorem [Thick-thin decomposition]: $M$ is diffeomorphic to the interior of a compact manifold $N$ with $\partial N$ that consist of a disjoint union of tori $T_1, \ldots, T_c$.
Fix the notation $\mathring{N}=M$.
One operation that you can perform on $N$ is the Dehn Filling: choose a boundary component $T_i$, choose a diffeomorphism $f_i \colon \partial(D^2 \times S^1) \to T_i$ and consider the new manifold
$$ N(f_i) = N \cup_{f_i} D^2 \times S^1$$
obtained by gluing a solid torus to the chosen boundary component using the chosen diffeomorphism. Through this operation, you fill one boundary component with a solid torus.
The manifold that you obtain depends on the diffeomorphism $f_i$ that you choose. Suppose that you perform Dehn Filling on every boundary component of $N$ with functions $f_1,\ldots, f_c$. Call the result of these Dehn Fillings $N(f_1,\ldots, f_c)$. It is a closed manifold. Now, there are two results that can help you. These can be found in the notes of Thurston, but I will give you another citation that may help you.
Theorem [1, Theorem 15.1.1, Hyperbolic Dehn filling, very basic version]: There are functions $f_1,\ldots, f_c$ such that $N(f_1,\ldots, f_c)$ admits a hyperbolic metric.
Theorem [1, Theorem 15.4.1]: Suppose that $N(f_1,\ldots, f_c)$ admits a hyperbolic metric. Then $vol(N(f_1,\ldots, f_c)) < vol(M)$.
Using all these results, it is easy to show that a cusped hyperbolic manifold with volume less than the smallest volume of a closed hyperbolic manifold cannot exist: in this case, you could Dehn Fill it and obtain a closed manifold with even smaller volume, contradicting the hypothesis.
[1]: Martelli Bruno, An Introduction to Geometric Topology.
