If I have a plane $\mathbb{R}^2,$ and I take out $n$ points, I can picture it wlog as the union of two planes, one without a point and one without $n-1$ points, and these two halves of the plane intersect at a line.
Since everything is open and path connected I can use Van Kampen to obtain that the $\pi_1$ of the plane minus $n$ points is the free product of the $\pi_1$ of the plane minus one point and the $\pi_1$ of the plane minus $n-1$ points.
Iterating this we get that the fundamental group of the plane minus $n$ points is the free product of the $\pi_1$ of the plane minus a point $n$ times.
But the plane minus a point can be retracted to a circle, hence what we have is the free product of $\mathbb{Z}$ with itself $n$ times.
I am posting this to check if my reasoning is correct.