This is a variant of the orchardplanting problem that uses circles instead of straight lines.
The maximal number of 3point circles passing through n points on a plane is binomial(n,3). Given an arrangement of n points in general position, any choice of three points defines a circle.  Peter Kagey, Oct 05 2020
Paul Panzer provides upper and lower bounds:
a(n) <= floor[n*(n1)*(n2)/24].
a(n) >= 2 + n*[(n2)*(n2)+4]/32 for n == 0 mod 4 and n >= 8.
a(n) >= 2 + (n1)*[(n1)*(n5)+16]/32 for n == 1 mod 4 and n >= 9.
a(n) >= 2 + n*(n2)*(n2)/32 for n == 2 mod 4 and n >= 10.
a(n) >= 2 + (n1)*[(n3)*(n3)+16]/32 for n == 3 mod 4 and n >= 11.
