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][1] establishing a conjecture of Erdos 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/Erdos-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. 

[1]: http://archive.numdam.org/ARCHIVE/JTNB/JTNB_1990__2_1/JTNB_1990__2_1_13_0/JTNB_1990__2_1_13_0.pdf