My favorite example from algebraic topology is Rene Thom's work on cobordism theory. The problem of classifying manifolds up to cobordism looks totally intractable at first glance. In low dimensions ($0,1,2$), it is easy, because manifolds of these dimensions are completely known. With hard manual labor, one can maybe treat dimensions 3 and 4. But in higher dimensions, there is no chance to proceed by geometric methods.