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.

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 topology generated by 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.