I particularly like the derivation why arguments to sin etc. must be dimensionless.
One aspect where dimensions have helped me in exam situations was this: Given three (or more) units, dimension analysis often lets you derive the mathematical formula of the relationship of their quantities, e.g.
work = force * length
because
joule = newton * meter
One might envision a dimension-aware Djinn that can derive such formulas given the units.
How does dimensional deal with integrals and derivatives? Or is that subsumed by ordinary multiplication and division? Could it make sense to add Newton’s fluxions (infinitesimals) as an additional basic dimension? This would allow the scientific Haskeller to express differential equations in a type-safe way.