Given a surface, the fine $k$-curve graph of the surface is a graph whose vertices are simple closed essential curves and whose edges connect curves that intersect in at most $k$ points. We note that the fine $k$-curve graph is hyperbolic for all $k$ and, for $k\geq 2,$ show that it contains as induced subgraphs all countable graphs. We also show that the direct limit of this family of graphs, which we call the finitary curve graph, has diameter 2, has a contractible flag complex, contains every countable graph as an induced subgraph, and has as its automorphism group the homeomorphism group of the surface. Finally, we explore some finite graphs that are not induced subgraphs of fine curve graphs.