Consider the real line R $\mathbb R$ and $C^1_0$ , the ring of germs of continuously differentiable functions at zero. Now take the ideal M $M$ of germs vanishing at zero. The Zariski cotangent space $M/M^2$ has dimension the continuum (because the classes of $x^{1+\epsilon}$ are linearly independant in the quotient for $0<\epsilon <1$ ). Hence the Zariski tangent space of the real line , i.e. the dual of $M/M^2$ , has dimension $2^{continuum}$ . Some geometers might think this is a bit large for the real line.
This result is essentially exercise 13 of Chapter 3 of Spivak's Differential Geometry , Volume I.
