How Good Are the Upper Limits on Heavy Elements? There are between 90 and 254 stable nuclei all the way up to element number 82. In discussions and graphs about big bang nucleosynthesis nothing above lithium is even mentioned. It's a pretty safe bet that none of the heavier elements have been observed in reasonably primordial gases, since the implications would be profound enough to require mention in even popular level books about the big bang.

That said, has anyone bothered to perform searches for all of the heavier elements? As in, do we have explicit experimental upper limits on the concentrations of some or all of these heavier elements, or just a more qualitative, "No evidence seen for anything else"?
Good question! Normally, the lack of heavy elements is taken as an indicator that something was made from (close-to-primordial) gas.

So an answer could be to ask what are the lowest abundances ever measured with respect to hydrogen.

I am not completely up to date with the current record holders, but stars with iron abundances that are 5 orders of magnitude less than in the Sun have been found (Norris et al 2013). This corresponds to A(Fe)$\sim 2$ on the usual logarithmic scale where hydrogen has A(H)=12.

The constraints on other iron-peak elements are similar. Alpha elements like O, Mg are usually enhanced in very metal poor stars, so the constraints are an order of magnitude higher.