Any more progress?
I've extended my sequence to n=10 and have found a nice pattern such that I can probably make the maximum for any n, by starting with a regular polygon and taking some of its diagonals, then making adjustments to avoid multiple intersections. The method alternates between even and odd n, and the formula for the number of regions reflects that, though it can be written as a single formula using the ceil function. All I lack is proof that my method really accomplishes the goal (as if that were trivial).
I'm eager to help if you have been working on it. Until then, all I have are these hints.