Consider the hyperconnected space $(X, \tau).$
The poset $P=\{\tau': \tau\subseteq \tau', (X, \tau')$ is a hyperconnected topological space$ \}$ ordered by inclusin satisfies the requirement of the Zorn's lemma (any increasing chain has an upper bound, namely the union of the elements of the chain), so it has a maximal element, call it $\tau'.$ Then $(X, \tau')$ is a maximally hyperconnected space.