Consider a single-pile NIM variant, played under standard (not misere) objective, with the rule that you may remove any *prime* number from the pile. The winning positions of this game are all numbers $a_n$ where $a_0 = 0$ and for $A \equiv \{a_n\}$ consists of all numbers which are *not* of the form $p+a_k : p \text{ prime}, k<n$.
(This sequence appears, to 10,000 terms, in OEIS as A025043.)

What is the asymptotic density (and if that is zero, bounds on a counting function) of winning positions?

Knowing the counting function of primes, one can frame a probabilistic argument about the likelihood that a given number $k \in A$ by solving an integral equation that looks something like $$ D(x) = 1 - \int_1^u\frac{D(u)}{\log u} du $$ (and you don't even have to solve the equation, just determine the asymptotic behavior of the solution). But that, of course, does not prove anything, since the primes could easily conspire to make the distribution different than that naive probability argument would suggest.