Having trouble finding examples of sets of functions that do NOT form a vector space In this question someone asks about showing that the set of all functions of the form $y(t) = c_1\cos\omega t + c_2\sin\omega t$ is a vector space. But doesn't literally any set of functions of the form $y(t) = c_1f(t) + c_2g(t) + \ldots$ form a vector space? After all, there will always be a zero element (coefficients = 0) and an additive inverse (coefficients of opposite sign), and trivially scaling or adding two $y(t)$ will yield another function of the same form.
So what is an example of a set of functions that do NOT form a vector space? The most common pedagogic example I've seen is unsatisfyingly contrived: the set of all polynomials of degree N. This is explained to not form a vector space because the zero element is not of degree N. However technically the zero element is still of the form $c_1 + c_2x + \ldots + c_Nx^N$, so I'm not sure how comfortable I am with this example. 

Take your favourite differential equation, as long as it's not a homogeneous linear one. Then its solution set isn't a vector space under the usual definitions for functions of scaling and addition. Surely that's not a contrived example.
For the sake of physical examples, note that the linearity of quantum mechanics and electromagnetism is starkly at odds with the nonlinearity of general relativity and nuclear interactions. The Higgs field satisfies a nonlinear differential equation. While electromagnetism is linear, it's inhomogeneous with a source. Fluid mechanics is nonlinear too.
