  A set of positive integers is primitive (or 1-primitive) if no member divides
another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ for
$n$ ranging over a primitive set $A$ is universally bounded over all choices
for $A$. In 1988 he asked if this universal bound is attained by the set of
prime numbers. One source of difficulty in this conjecture is that $\sum
n^{-\lambda}$ over a primitive set is maximized by the primes if and only if
$\lambda$ is at least the critical exponent $\tau_1 \approx 1.14$.
  A set is $k$-primitive if no member divides any product of up to $k$ other
distinct members. One may similarly consider the critical exponent $\tau_k$ for
which the primes are maximal among $k$-primitive sets. In recent work the
authors showed that $\tau_2 < 0.8$, which directly implies the Erd\H{o}s
conjecture for 2-primitive sets. In this article we study the limiting behavior
of the critical exponent, proving that $\tau_k$ tends to zero as $k\to\infty$.
