What's nonsensical about this definition of the order of a meromorphic function?

Yes, so Georges is correct. The definition is flawed. The correct definition is that that the order of $f$ is the largest integer $k$ so that you can express $f = g^k h$ in the local ring. There is no reason to expect $h$ to be a unit. (Great question, by the way. I'm embarrassed it took me so long.) See the various examples that @JulianRosen and I offered in the comments.

Does ONLY the ellipse have these properties?

If $f:\mathbb R^2 \to \mathbb R$ continuous on straight lines and $f(\text{compact})= \text{compact}$, then $f$ continuous?

Zelda - Oracle of Ages tiles rooms puzzles: how can you prove if there are/aren't solutions?

Prove $\prod_{i=1}^n \frac{3K_i+2}{2K_i+1}$ can never be a power of $2$

For what $x$ is there some $k$ such that $x\uparrow\uparrow k$ is an integer?

Counting minimum elements needed such that their sum covers the whole finite space.

Min $P=\frac{x^5y}{x^2+1} +\frac{y^5z}{y^2+1} +\frac{z^5x}{z^2+1}$

Can the idea of a 'function of a variable' be made rigorous?

Showing the Fibonacci inequality $f_1^{f_1}f_2^{f_2}f_3^{f_3}\cdots f_n^{f_n}\leq f_1!f_2!f_3!\cdots f_n!\;e^{({f_{n+2}-n-1)}}$ without induction.

What should I study to understand Milnor's exotic 7-spheres?

Why the mapping $1/\bar{z}$ look like the field lines of an electric dipole?

Square differential forms in cohomology