Background: I'm doing worldbuilding for my D&D world, and I want one location to be an "infinite city": finite circumference from the outside, but as you move toward the "center" there's always more room and space to go.
A specific detail I'd like is for "ring roads" circumnavigating the space inside to exist, and have larger circumference as you go further "in".
I think this can be described by saying that the region of space has a highly-curved hyperbolic geometry. Is this correct? If not, is there another way of describing the space that would have the effect I want? It's okay if the resulting geometry wouldn't be suitable for physics as we know it, I'm fine with hand-waving "it's magic!" to handle that; I'd just like something I can think about and possibly obtain some non-obvious interesting details from, and get a consistent description out of.
Such examples are easy to construct indeed using classical hyperbolic geometry; however, you need to know some differential geometry.
Start with the upper half-plane ${\mathbb H}^2$ in ${\mathbb R}^2$ equipped with the hyperbolic metric $$ ds^2= y^{-2}( dx^2 + dy^2). $$
Next, consider the rectangular (in the Euclidean sense) region $$ Q=\{(x,y): 0\le x\le 2\pi, 0< y\le 1\} \subset {\mathbb H}^2.$$
Identify the vertical boundary intervals of $Q$ via horizontal translations: $$ (0,y)\sim (2\pi,y), 0< y\le 1. $$ The quotient space $A$ is diffeomorphic to the half-open annulus (a closed disk with the center removed), $$ A\cong S^1\times (0,1]\cong D =\{w\in {\mathbb C}: 0< |w|\le \frac{1}{e}\} $$ The hyperbolic metric $ds^2$ projects to a Riemannian metric on $A$ (since horizontal translations are isometries of the hyperbolic metric). You can realize this diffeomorphism via the map $$ z=x+iy\mapsto w=\exp( i x - y^{-1}), z\in Q, w\in D. $$ Your concentric roads $C_r$ in $D$ are the Euclidean circles $|w|=r$. But the hyperbolic lengths $L_r$ of such roads are equal to $$ \int_{0}^{2\pi} \frac{dx}{y}= \frac{2\pi}{y}, $$ where $$ r= \exp(-y^{-1}), y= \frac{-1}{\ln(r)}. $$ Hence, as $r\to 0$, $L_r\to\infty$. The distance between $C_r$ and the exterior road $C_{1/e}$ is constant, equal $-\ln(y)$, where $y= \frac{-1}{\ln(r)}$. The distance from any point of $D$ to the center is infinite: If $w\in D$ corresponds to $z=x+iy$ then the distance from $w$ to $0$ equals $$ \int_{0}^y \frac{dt}{t}= - \lim_{t\to 0} \ln(t)= + \infty. $$ The total area of $D$ is infinite as well (you compute it using the same integral).