  This paper studies how long it takes the orbit of the chaos game to reach a
certain density inside the attractor of a strictly contracting iterated
function system of which we only assume that its lower dimension is positive.
We show that the rate of growth of this cover time is determined by the
Minkowski dimension of the push-forward of the shift invariant measure with
exponential decay of correlations driving the chaos game. Moreover, we bound
the expected value of the cover time from above and below with multiplicative
logarithmic correction terms. As an application, for Bedford-McMullen carpets
we completely characterise the family of probability vectors which minimise the
Minkowski dimension of Bernoulli measures. Interestingly, these vectors have
not appeared in any other aspect of Bedford-McMullen carpets before.
