Does there exist a Riemannian manifold $(M,g)$ such that for every $S>0$ there is a geodesic unit ball $B\subset M$ with $\text{Vol}(B)>S$?
If I understand the Bishop-Gromov-inequality correctly, then this cannot happen for complete $(M,g)$. However I am mostly interested in the non-complete case.
My attempt to find statements about $\sup_B \text{Vol}(B)$ were not very succesful, because most people seem to be interested in lower bounds on manifolds with finite volume.
Let $(M,g)$ be the disjoint union of complete simply-connected $n$-dimensional Riemannian manifolds $X_i$ of curvature $-i^2$, where $i\in {\mathbb N}$ (each component of $M$ is, therefore, a rescaled copy of the hyperbolic $n$-space). Then if $B_i\subset X_i\subset M$ is the sequence of unit balls, we obtain $$ \lim_{i\to\infty} Vol(B_i)=\infty. $$ One can easily modify this example to make it connected (and, with more work, even contractible).