Hexagon Tessellations

[merlin2007]

A configuration is made of congruent regular hexagons, where each hexagon shares a side with another hexagon. What is the largest integer k, such that the figure cannot have k vertices?

I've been trying approaches where I count the number of vertices that can be added in a given configuration, but is there a more straight-forward approach? I feel like I'm using a a shovel to dig out a parking lot.

Thanks