Coupon collector problem for collecting set k times.

I'm closing some other questions as a duplicate of this one, so I'll collect the information that's been posted in various places in one place:

This problem was treated in the paper The double Dixie cup problem by Newman and Shepp. The expected collection time is

$$ n \log n + (m-1) n \log\log n + O(n)\;, $$

so, as Steven Stanicki noted, each additional set collected takes $n\log\log n$ time.

In Some new aspects of the coupon collector's problem, Myers and Wilf gave a simpler derivation of the result and a generating function in one variable.

The result is also derived in Section $5$ of The Coupon Collector’s Problem by Ferrante and Saltalamacchia, which treats various generalisations of the coupon collector's problem with various approaches.

It asymptotically almost surely takes $O(n \log n)$ time to collect $n$ cards, so for any constant number of sets it should take (a.a.s.) $O(n \log n)$ time as well, since at worst case you can imagine 'forgetting' to count duplicates until you have a new set. Therefore you can only hope to improve the constant, unless you are hoping to incorporate the number of sets ($k$) as a variable.