Graphs for maximum number of min cuts?
Aside from the cycle graph, I think a nontrivial class of graphs with $\Omega(n^2)$ 
distinct min cuts are cactuses. A cactus  is a connected graph in which every block (a maximal 2-connected subgraph) is a chordless cycle. 

EDIT: In case the considered cactus has $n$ nodes and $b$ blocks of order $n_1, n_2, \ldots, n_b$, the number of min cuts is $\Omega\left(\sum_{i=1}^b n_i^2\right)$. 
As $\sum_{i=1}^b n_i^2 \ge \frac{1}{b} n^2$, the number of min cuts is $\Omega(n^2)$, 
provided the number of blocks $b$ is bounded.