A torsion free connection that is not metric compatible

On $\mathbb{R}$ with $g=d{x}^2$, prescribe ${\mathrm{\Gamma }}^1{}_{11}=1$, equivalently the matrix $dx$ in tangent frame ${\mathrm{\partial }}_x$. This gives a smooth affine connection by [F1], with ${\mathrm{\nabla }}_{f{\mathrm{\partial }}_x}(u{\mathrm{\partial }}_x)=f(u^{\mathrm{\prime }}+u){\mathrm{\partial }}_x$. Its sole lower-index pair is symmetric, so [F2] gives torsion zero.

Set $X=Y=Z={\mathrm{\partial }}_x$. The left side in [F3] is ${\mathrm{\partial }}_{x}1=0$, while its right side is $1+1=2$, so compatibility with $d{x}^2$ fails. This does not assert failure for every metric: with $\tilde{g}=e^{2x}d{x}^2$, the one-dimensional compatibility equation is ${\mathrm{\partial }}_x{e}^{2x}=2{\mathrm{\Gamma }}^1{}_{11}{e}^{2x}$, which holds. For arbitrary local multiples the additional derivatives of their coefficients match by the scalar product rule, so this last test indeed gives compatibility with $\tilde{g}$.