What problem in pure mathematics required solution techniques from the widest range of math sub-disciplines?
The proof of the Ramanujan conjecture by Deligne. It uses:
algebraic geometry
topology
representation theory
commutative algebra
complex analysis
I'm not qualified to certify optimality, but I've always thought that the Mostow rigidity theorem is a good candidate. The theorem says that every isomorphism between the fundamental groups of two finite volume hyperbolic manifolds of dimension at least 3 is induced by a unique isometry. Mostow's original proof (for the compact case) used:
- Riemannian geometry
- Conformal geometry
- Geometric group theory
- Representation theory
- Ergodic theory
- A dash of number theory
For generalizations to symmetric spaces you need algebraic geometry and more serious number theory as well.
One relatively recent result that comes to mind is the Kadison-Singer problem, which was originally formulated in 1959 as a question in $C^*$-algebra theory, but was successively reduced to more tractable and accessible questions in other fields by several mathematicians (including the MathOverflow member Nik Weaver). It was solved in 2013 by the computer scientists Marcus, Spielman and Srivastava using properties of random polynomials.