This recent question asks for a set of forms (binary quadratic) representing all primes. Set of quadratic forms that represents all primes
When the question was asked on MSE last month
I made the claim that no finite set of positive binary forms would suffice. This still seems right to me, but I lack a proof or any reference. The subject is traditional, I would guess there is a mention in, say Dickson's History, which I do have. I will check.
Let's see, this will take some time, but there is no problem writing a Manjul Bhargava style "truant" program, begin with $x^2 + y^2,$ prime $3$ missing says add $x^2 + 2 y^2,$ then $7$ missing says add $x^2 + xy + 2 y^2,$ and so on. Eventually I would expect to see some non-principal forms as the smallest absolute discriminant form.