Are there any other interesting topologies on $\mathrm{Spec} (A)$ other than Zariski's?

As far as I'm aware there is no other interesting topology on $\mathrm{Spec} (A)$. However:

1) There are the so called "Grothendieck topologies" on any $\mathrm{Spec} (A)$ or more generally on an arbitrary scheme.
They are generalizations of the usual notion of a topology and compensate for the coarseness of the Zariski topology, in that they allow results analogous to those in algebraic topology: Gysin sequence, Künneth formula, Poincaré duality, Lefschetz fixed point formula,etc.
They were dreamed of by Weil and constructed by Grothendieck (with the help of Mike Artin and others), spurred by an insight of Serre's.
Their most spectacular achievement was the complete solution of the Weil conjectures by Grothendieck and Deligne.

2) For schemes $X$ of finite type over $\mathbb C$, for example $\mathrm{Spec} (A)$ with $A$ a finitely generated $\mathbb C$-algebra, Serre showed how to give the set $X^{cl}$ of closed points of $X$ a topology and a structure sheaf such that $X^{cl}$ becomes a complex analytic space.
There is then an exciting interplay (foreshadowed by Riemann, Chow and others) between the scheme $X$ and the analytic space $X^{cl}$, in which algebra, algebraic topology and analysis join forces, to our greatest delight.
Hodge theory is an egregious example of that alliance.

There is a generalization of the notion of topology called a Grothendieck topology. Grothendieck has defined many examples of Grothendieck topolgies in the category of schemes for example the Etale topology which is one notion used in the proof of Weil conjectures.

https://en.wikipedia.org/wiki/Grothendieck_topology