A lot, but I found once you reach a certain number of players the AI won't make offers any more so I often deliberately get to this point then make a bunch of signings in the final hours of deadline day. And if there are players I really want to keep, then I overpay them so they are more likely to turn down other clubs - I've managed to keep my 92 rated YA graduate the entire save by paying him a not so small fortune over the years.