Given $n$ polynomials $f_i(x) = a_i x^2+ b_ix+c_i$, where $a_i, b_i, c_i \in \mathbb{R}$. And assume that no two polynomials are identical. The minimum number of intersections is 0 since one can let $\forall i, a_i = a, b_i=b$. And then it's the same polynomial shifted in parallel.
How about the maximum number of intersections?
By intersection, I mean points where at least two $f_i$ vanish.
Let $f_i(x)=ix^2-\exp(i)$. $f_i$ and $f_j$ meet at $$x_{ij}^2=\frac{\exp(i)-\exp(j)}{i-j}$$
If any $x_{ij}=x_{kl}$ then the transcendental $e$ would satisfy a polynomial with integer coefficients.