If it's ok with you, I'm going to list my favorite exhibits on the basis of species or taxonomic group.
Elephants
Dallas Zoo - Giants of the Savannah - ten elephants on five acres, half an acre per elephant, makes this one of the biggest elephant exhibits in the US. They have lots of space and can mingle with other African wildlife like guinea fowl, giraffes, and zebras.
Apes
Bronx Zoo - Congo Gorilla Forest - A 6.5-acre themed around the Congo Rainforest. It has a large space for a troop of western lowland gorillas with grass, hills, and plants. The area is also home to okapis, mandrills, birds, and herptiles. Worth the separate admission.
Giraffes
Columbus Zoo - Heart of Africa - Alongside zebras, antelopes, and birds, they live in a 23-acre grassy plain in good sized groups. Guests can see them from, among other viewing areas, the food court through a big glass window. Great photo ops!
Reptiles
Tie between Fort Worth Zoo's Museum of Living Art (MOLA) and St. Louis Zoo Herpetarium. Both are large buildings with a diverse array of herptiles. MOLA also has some birds, tamarins, and lemurs. STL's reptile house has two levels. Highlights include Lake Titicaca frogs (both), saltwater crocodile (FW), Aldabra tortoise (STL), and tomistoma (STL).
Penguins
Tie between Detroit Zoo's Polk Penguin Conservation Center and St. Louis Zoo's Penguin & Puffin Coast. Detroit's is themed around Antarctic exploration and has moving water and cool graphic signs. St. Louis has an outdoor Humboldt penguin exhibit and a larger chilled indoor exhibit for gentoo, king, and rochopper penguins. Penguins on all sides of you, it is the first walkthrough penguin exhibit in America. It is followed by a puffin exhibit. The penguins and puffins are so close, your could (but, of course, are not allowed to) touch them.
Cats
Lion: Franklin Park Zoo - Kalahari Kingdom - classic moat exhibit. Pretty big space with two males. Shady and lots to do.
Tiger: Bronx Zoo - Tiger Mountain. Large and lush space for Amur tigers. Conservation messages surround the visitor.
Misc. cats: Snow leopard - Bronx Zoo - Himalayan Highlands - lots of rocks and vertical space.
Ocelot - Buffalo Zoo - Rainforest Falls - Spacious, especially vertically.
Bears
Polar: Detroit Zoo's Arctic Ring of Life or Buffalo Zoo's Arctic Edge. Both have large dynamic spaces for their bears, but Detroit is the largest of the two. ARoL is also home to seals.
Brown or grizzly: Akron Zoo's Grizzly ridge or Bronx Zoo's Big Bears. Both have large spaces with water, plants, and rocks for grizzly and brown bears, respectively. Bronx has a large rock feature for their bears to climb on.
Misc. species: Queens Zoo - spectacled bear. Big space with climbing opportunities, hills, and water.
National Zoo - giant panda. Very hilly, perfect for these mountain-dwelling bears.
Marine mammals
Cetaceans: Shedd Aquarium - Abbot Oceanarium - large naturalistic spaces for Pacific white-sided dolphins and beluga whales.
Pinnipeds: Tie between Bronx Zoo's Sea Lion Pool and St. Louis Zoo's Sea ion Sound. Both have large spaces. STL is more dynamic and has an underwater tunnel, but clear blue water (not good for their eyes), and they share with harbor seals. Bronx has no underwater viewing, but is also large and provides some privacy.