If StPHawk or any of those guys are lurking, feel free to chime in, but I would say Minnesota is Iowa's biggest rival. I could see an argument for Iowa State, but beyond that, I don't know that Wisconsin or Illinois fits the bill.
Iowa and Wisconsin play for that ridiculous trophy (the Heartland Trophy) that wasn't created until 2004. I'm not saying a trophy makes a rivalry, but artificially sticking a trophy on a series that had been played 79 times already, dating back to 1894, seemed kinda weird and mostly just to try and build up regional/national attention for the rivalry, since it's now a "trophy game." I will say it's a balanced series though. 41-41-2 all time, with Wisconin up 53 points cumulatively (1,471 - 1,418) over the years.
Iowa and Illinois I'd struggle to categorize as a top rivalry simply because they don't play every year. It's just like the Minnesota/Michigan rivalry, minus the tradition behind the Little Brown Jug. I would bet if you asked Iowa fans, Illinois would rank (at best) third on their list of rivals, and in turn, Illinois would say the same about Iowa (behind Mizzou and Indiana).