In example 13.31 (page 343) of Introduction to Smooth Manifolds, Lee uses the musical isomorphisms to calculate the gradient in polar coordinates.
He obtains: $$\text{grad} f = \frac{\partial f}{\partial r} \frac{\partial}{\partial r} + \frac{1}{r^2} \frac{\partial f}{\partial \theta} \frac{\partial}{\partial \theta}. $$
The $1/r^2$ terms differs from every other expression for the gradient in polar coordinates I have seen. In every other version, it is a $1/r$ term.
For example: How to obtain the gradient in polar coordinates and https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates
I don't see any errors in Lee's derivation. What am I missing?
The coefficient is different, but the expression is still correct.
The expressions that you see in the other versions are written with respect to $\hat{e}_\theta$, which is, by definition, the unit vector along the $\theta$ direction. Here, "unit vector" means unitary with respect to the Euclidean metric. However, recall that the Euclidean metric is written in polar coordinates as $$g=dr^2+r^2d\theta^2$$ so the norm of $\frac{\partial}{\partial \theta}$ is $r$ and not $1$. Hence $\frac{\partial}{\partial \theta}=r\hat{e}_\theta$.