Background
Conventionally, the square-cube law maintains that as the length scale of a regular shape increases, the surface area-to-volume ratio will dimish to zero. A cube with side length $l$, for example:
$$\lim_{l\to\infty} \frac{SA}{V} = \lim_{l\to\infty} \frac{6l^2}{l^3}=0$$
Interestingly, there exists at least one shape in which the opposite is true, where the surface area-to-volume ratio is unbounded. This shape is Gabriel's Horn. It is the shape defined by revolving the function $y=\frac{1}{x}$ about the x-axis, taken from $x=1$ to some end point.
For Gabriel's Horn, we have:
$$\lim_{x\to\infty} \frac{SA}{V} = \lim_{x\to\infty} \frac{\int\limits_{1}^{x} 2\pi \frac{1}{x}\sqrt{1 + \frac{1}{x^4}} dx}{\int\limits_{1}^{x} \pi \frac{1}{x^2} dx}$$
For the numerator, $\frac{1}{x} \sqrt{1 + \frac{1}{x^4}} > \frac{1}{x}$ because the term $\sqrt{1 + \frac{1}{x^4}} > 1$. Thus, if we rewrite the integral to make it easier to solve:
$$\lim_{x\to\infty} \frac{2\pi \int\limits_{1}^{x} \frac{1}{x} dx}{\pi \int\limits_{1}^{x} \frac{1}{x^2} dx} = \lim_{x\to\infty} \frac{2\pi (\ln{x} - \ln{1})}{\pi (\frac{1}{1} - \frac{1}{x})}=\frac{\infty}{1}=\infty$$
Because the actual numerator is greater than the substituted term, Gabriel's Horn has an infinite surface area-to-volume ratio.
Question
Is there a shape such that the surface area-to-volume ratio, as a characteristic length increases, approaches 1?
Relevancy
Perhaps this is of mathematical interest on its own, but it also has important applications, especially within engineering (my field of expertise). Heatsinks, for example, have mass proprotional to their volume but their cooling efficacy is roughly proportional to their surface area. In this case, a high ratio is preferred. In certain chemical reactions, heat is produced by the reaction. Heat also tends to catalyze reactions, so a low ratio would result in a faster reaction since less heat is lost from the exposed surface area and more is generated from the increased volume. There are times when a system is difficult to scale because of how the square-cube law affects performance metrics in a disproportionate way, so a ratio of 1 would allow for infinite scalability with proportional cost and performance.
My knowledge of mathematics is fairly limited, but I believe this also has applications to fractals (which in turn have applications to antennae), which often behave similarly to Gabriel's Horn in how their ratios behave.