Consider the irregular quadrilateral tiling of the Euclidean plane depicted by the log-log coordinate grid:
I'm wondering if in the Hyperbolic plane exist some analog of this kind of tiling where the polygons are of the same number of sides, but the sizes/lengths change according to some prescription?
In the upper half plane model, take the line $\{y=ai-\frac{1}{2}\mid a\in\mathbb{R}^+\}$ and $\{y=ai+\frac{1}{2}\mid a\in\mathbb{R}^+\}$ and some sequence of circular arcs between these lines whose center is at the origin, and whose radii decrease to zero at whatever rate you like. Now take all of these lines and arcs and translate it by all the integers horizontally. This is a tiling of the hyperbolic plane by hyperbolic quadrilaterals.
There are of course many ways to alter this, and it's hard to know what properties you exactly want this tiling to have. Please be more specific if you want it to have a particular property for which you are not sure if it actually exists or not.