In Sid Sackson's classic book A Gamut of Games, he introduces a game that he calls "Hold That Line." Briefly, it is an impartial pencil-and-paper game played on a finite grid of dots. The first player connects two dots with a horizontal, vertical, or diagonal line; thereafter, each player extends the given piecewise-linear curve at one end, making sure to keep it self-avoiding.
Analyzing this game looks like a nice project in combinatorial game theory, suitable for undergraduates, but has it already been studied? A quick search only turned up a brief discussion in Alan Lipp's book The Play's the Thing.
EDIT added March 2019: This question (among others) was raised by Jim Henle in his Spring 2019 Mathematical Intelligencer article, "Mathematical Treasures from Sid Sackson," but without any analysis (other than the observation that the normal form version has a Tweedledum-Tweedledee winning strategy for the first player). Henle has a webpage for recording comments about his article, but as of this writing, there are no comments about Hold That Line.