Let E(n, m) be the expected maximum number of half pills after starting with n whole pills and m half pills. The answer we're looking for is E(n, 0). I tried to solve for it in terms of E(n-1, 0).
This is as far as I got:
E(n, 0) = 2/(n^2 (n-2)!) + (1 - (n-1)(n-2)/n^2)E(n-1, 0) + (n-1)(n-2)/n^2...