I'm looking for smooth (infinitely differentiable everywhere) functions (curves) $\mathbb{R}\rightarrow\mathbb{R}^d$ that are approximately space-filling, i.e. scaling allows the curve to eventually get arbitrarily close to all points in $\mathbb{R}^d$.
An intuitive example for $\mathbb{R}\rightarrow\mathbb{R}^2$ would be the Archimedean Spiral, e.g.:
Example function:
$$ \mathrm{f}(t)= \rho\cdot \begin{pmatrix} \cos(t) \cdot t \\ \sin(t) \cdot t \end{pmatrix} $$
As $\rho$ approaches zero, the spiral will eventually get arbitrarily close to every point in $\mathbb{R}^2$.
It would also be great if the computational complexity of calculating such a function only increases linearly with the dimension $d$.
I have it!
The idea is to multiply a parameter $t^{\frac1{d+1}}$ by various periodic functions of arguments that depend non-linearly on $t$ and contain coefficients which are not rationally related. Then eventually any spot in space gets approached arbitrarily closely, yet in more than 2 dimensions the curve is non-intersecting.
The example I have in mind is something like: $$ x = \sqrt[4]{t} \sin \left( t + \frac{t^2}{\sqrt{2}} \right) \\ y = \sqrt[4]{t} \cos \left( \sqrt{3}t + t^2 \right) \\ z = \sqrt[4]{t} \sin \left( \pi t^2 \right) $$ The low-radius area gets filled fairly thoroughly because the curve keeps zipping through it when all the periodic functions coincide near zero. And that statement appears to be scale independent.
The reason that $t^{\frac{1}{d+1}}$ is used is so that the (hyper)volume being traversed grows more slowly than the length of curve within that volume. That may or may not be a necessity.
I would have difficulty with a rigorous proof for any given fixed $\epsilon$ and any point $\vec{x}$ there exists some $\delta(\epsilon,\vec{x})$ such that the curve intersects an $\epsilon$-ball about $\vec{x}$ for some $t < \delta$ but for generic (non-special) choices of the coefficients in the periodic functions I would be shocked to learn that "holes" exist.