Smallest value of largest angle in finite planar configurations
The key bound is $(1 - 1/n) \pi$, due to Erdős and Szekeres:
The above is an excerpt from this paper:
The Erdős-Szekeres result is in their 1961 paper, "On some extremum problems in elementary geometry.", Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Math. 3-4, 53-62 (1961) (PDF download link).
Consider an exagon obtained from an equilateral triangle by cutting three small equilateral triangles from its vertices. Consider the configuration of the $6$ vertices of the exagon, plus the center of the initial equilateral triangle. It seems to me that with these $7$ points one can't do angles larger than $2\pi/3+\epsilon$.