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Indeed such problems have been studied before. First note that if $p_1^{k_1} p_2^{k_2}\cdots \le x$ then $p_1p_2\cdots \le x$. Thus the champions will always be in the form of a number of $1$'s. If you have $r$ ones, then you're asking for the number of square-free integers up to $x$ with exactly $r$ prime factors. This has been widely investigated. For example, Balazard establishing a conjecture of Erdős showed that for each large $x$ this sequence is unimodal in $r$ (see Theorem E on page 24), and (as one can see by Hardy-Ramanujan/Erdős-Kac) attains its maximum for $r$ close to $\log \log x$. Balazard's paper will have more references -- in particular to work of Hildebrand and Tenenebaum giving asymptotics in wide ranges.