  Wiener index, defined as the sum of distances between all unordered pairs of
vertices, is one of the most popular molecular descriptors. It is well known
that among 2-vertex connected graphs on $n\ge 3$ vertices, the cycle $C_n$
attains the maximum value of Wiener index. We show that the second maximum
graph is obtained from $C_n$ by introducing a new edge that connects two
vertices at distance two on the cycle if $n\ne 6$. If $n\ge 11$, the third
maximum graph is obtained from a $4$-cycle by connecting opposite vertices by a
path of length $n-3$. We completely describe also the situation for $n\le 10$.
