The maximum possible toughness among graphs with \(n\) vertices and \(m\) edges is considered for \(m \geq \lceil n^2/4 \rceil\). We thus extend results known for \(m \geq n\lfloor n/3 \rfloor\). When \(n\) is even, all of the values are determined. When \(n\) is odd, some values are determined, and the difficulties are discussed, leaving open questions.