  Every graphon defines a random graph on any given number $n$ of vertices. It
was known that the graphon is random-free if and only if the entropy of this
random graph is subquadratic. We prove that for random-free graphons, this
entropy can grow as fast as any subquadratic function. However, if the graphon
belongs to the closure of a random-free graph property, then the entropy is
$O(n \log n)$. We also give a simple construction of a non-stepfunction
random-free graphon for which this entropy is linear, refuting a conjecture of
Janson.
