Smooth curves of finite length have a Hausdorff dimension of 1. How about smooth but non-rectifiable (i.e. infinite-length) curves? Are they also of Hausdorff dimension 1, or does it depend on the particular curve we are considering?
For example, consider the elementary curve $y = x^2$ in the closed right half plane. As $x \rightarrow \infty$, its arc length also grows approximately quadratically, compared to a straight line whose arc length grows linearly. Does this award our curve a Hausdorff dimension greater than one? If not, is there an established characteristic number that differentiates between such non-rectifiable curves?
Hausdorff dimension is countably stable meaning that the dimension of a countable union $\bigcup E_n$ is equal to $\sup_n \dim E_n$.
The graph of a $C^1$-smooth function $f:\mathbb R\to \mathbb R$ can be written as the union of the graphs of the restrictions $f_{[-n,n]}$. For each $n$, the graph of $f_{[-n,n]}$ is rectifiable, being a Lipschitz image of line segment $[-n,n]$. Therefore, this graph has Hausdorff dimension $1$. By countable stability, the graph of $f$ has Hausdorff dimension $1$.
To obtain graphs with Hausdorff dimension greater than $1$, one has to consider very nonsmooth functions such as the Weierstrass funtction.
You may want to consider (i) total curvature; (ii) the upper/lower exponents of curvature growth/decay; (iii) the upper/lower exponents of volume (length) growth... I could come up with a few more, but at some point the question shifts to: what is your goal, what do you want to use these numbers for? Coming up with definitions aimlessly is neither fun nor useful.