I'm not sure what Lang had in mind with "unnecessary double dualization", but here's an example that occurred to me in ancient times when I was trying to understand differential geometry better. Some (many? most?) people define tangent vectors to a manifold to be certain derivations on the smooth functions, and they define the cotangent space to be the dual of the tangent space. So a cotangent vector is a function taking as arguments tangent vectors, which are themselves functions taking as arguments smooth functions. In this sense, a cotangent vector is a doubly-dualized function. But one can avoid these dualizations by defining a cotangent vector at a point $p$ to be an element of $m/(m^2)$ where $m$ is the maximal ideal in the ring of germs of smooth functions at $p$. In other words, $m$ consists of the smooth germs that vanish at $p$ and $m^2$ consists of those that vanish to second order, so the quotient is "first-order data about a germ at $p$, omitting the value (zero-order data) at $p$." That picture captures pretty well my intuition of what a cotangent vector should be. (I think that the $m/(m^2)$ definition is used more in algebraic geometry than in differential geometry.)