Gino's, Malnati's, and Uno's
Also, I need a good Italian Beef recommendation.
I think your "Holy Trinity"
choice is excellent, esp. since they will all be somewhat close to the "River North" area in which you'll be staying. I would say those three are the best deep dish pizzas in Chicago, followed closely by Pizano's (the State St. location is very close to where you will be staying, too, just a couple of blocks off the magnificent mile). Giordano's is good, too, but I wholeheartedly agree that their stuffed pizza is just a variant of a deep dish pizza. I have a slight preference for the original Uno's sister restaurant, Due's, so if Uno's is too crowded, suggest you walk the short block to Due's, which is slightly larger. And I know you're aware that the original Uno's pizza is vastly superior to the junk pizza sold at nationwide Uno's Chicago Grills that have some strange business rights to the Uno's name (but not the pizza recipe).
Mr. Beef and Al's Beef are great places, but my favorite is Portillo's and their restaurant will just be a few short blocks walk from the original Uno's and Due's. I love their beef/sausage combo. Matter of fact, most of these places will just be a few blocks walk for you, except Malnati's on Wells St. may be a short cab ride, but if you feel up to it, you can walk to it.
Have a great trip this summer. I plan to be in the area on a couple occasions this summer, too, and consume some of those great pizzas.