The authors address aspects of this general question: Given an infinite compact group \(G=\langle G,{\mathcal T}\rangle\), is there a pseudocompact group topology (hereafter: PGT) \(\mathcal V\) for \(G\) such that \({\mathcal V}\supseteq{\mathcal T}\) and \({\mathcal V}\neq {\mathcal T}\)? If so, can \(\mathcal V\) be chosen of maximal weight (that is, so that \(w(G,{\mathcal V})= 2^{| G|})\)? Among the statements proved are these: Given a pseudocompact group \(F\) with \(| F|>1\) and given \(\kappa>\omega\), for every cardinal \(\beta\) such that \(w(F)+ \kappa\leq \beta\leq w(F)+ 2^{2^ \kappa}\) the product group \(\langle F^ \kappa,{\mathcal U}\rangle\) admits a PGT \(\mathcal V\) of weight \(\beta\) such that \({\mathcal V}\supseteq{\mathcal U}\), \({\mathcal V}\neq {\mathcal U}\). Every compact Abelian group \(\langle G,{\mathcal T}\rangle\) of weight \(\alpha\geq \omega\) admits a PGT \(\mathcal V\) of weight \(2^{2^ \alpha}\). If \(\alpha=\omega\) the choice \({\mathcal V}\supseteq{\mathcal T}\) is impossible. If \(\alpha>\omega\) and \(\langle G,{\mathcal T}\rangle\) is connected or torsion, or if \(\text{cf}(\alpha)>\omega\) or \(\log\log 2^{2^ \alpha}\omega\), and if either \(\text{cf}(\alpha)>\omega\) or the connected component \(A\) of the center of \(G\) satisfies \(w(A)=\alpha\), then there is a PGT \(\mathcal V\) on \(G\) such that \(w(G,{\mathcal V})=2^{2^ \alpha}\) and \({\mathcal V}\supset {\mathcal T}\).