I believe that the Berlin Zoo (if you count the aquarium) has the most species of any zoo, Hell it may even be without the aquarium it has the most species.
Nopes, Shedd Aquarium is the most species rich zoos, with 1433 species according to the latest International Zoo Yearbook, Zoo Berlin + Aquarium has 1363 species. Not counting the aquarium building Berlin would have less than 500 species.
Overview of all zoos and aquaria with 500 or more species according to the International Zoo Yearbook (based on data ranging from 2015-2017)
Zoos
Plzen Zoo 1383 (I think this includes species on loan)
Zoo Berlin 1363
Wilhelma Stuttgart 1151
Wroclaw Zoo 1139
Moscow Zoo 1102
San Antonio Zoo 1002
Houston Zoo 849
Zoo Budapest 828
Memphis Zoo 824*
Zoo Cologne 809
Zoo Leipzig 804
Bronx Zoo NYC 801 (probably includes all WCS collections)
Tierpark Berlin 790
California Academy of Sciences, San Francisco 783
Novosibirsk Zoo 738
Tierpark Hellabrun, Munich 727
Tiergarten Schönbrunn, Vienna 711
Pittsburgh Zoo 709
ZSL London Zoo 698 (will be significantly reduced now)
Krasnoyark 690
Prague Zoo 690
Toledo Zoo 677
St. Petersburg Zoo 676
Pairi Daiza 665
Moody Gardens Galveston 661
Denver Zoo 619
San Diego Zoo 655 (probably includes Safaripark, as listed as San Diego Zoo Global)
Vogelpark Walsrode 654
Columbus Zoo 636
Saint Louis Zoo 588
Zoo Basel 572
Zoo Beauval 566
Diergaarde Blijdorp 566
Tierpark Hagenbeck, Hamburg 558
Point Defiance Zoo Tacoma 557
Zoo Antwerp 556
Fort Worth Zoo 551
Artis, Amsterdam 548
Zoo Miami 535
Zoo Warsaw 531
Riverbanks Zoo 517
Zoo Tallinn 514
Brookfield Zoo 509
Burgers' Zoo 504
Nagoya Zoo 501
Chester Zoo, 500
Aquaria
Shedd Aquarium 1433
New England Aquarium 855
Niigata City Aquarium 834
Aquamarine Fukushima 770
Monterey Bay Aquarium 769
Oarai Aquarium 755
Aquario di Genova 757
Okinawa Churaumi 753
National Aquarium Baltimore 736
Vancouver Aquarium 734
Moskvarium 659
Den Bla Planet 656
Kamogawa Sea World, 634
Port of Nagoya Aquarium 630
Tennessee Aquarium 613
Haus des Meeres, Vienna 574
Oceanografic, Valenci 573
Tokyo Sunshine Aquarium 567
Hong Kong Ocean Park 551
Wakayama Museum of Natural History, Kainan 534
Non-listed zoos include Oceanopolis Brest, Nausicaa, Toba, all Chinese zoos, Georgia Aquarium, Omaha Henry Doorly Zoo, Philadelphia, Dallas World Aquarium
Major zoos with 400-500 species include Toronto, Ostrava, Duisburg, Frankfurt, Wuppertal, Sosto, Africam Safari (Mexico), Bergen Aquarium, Oceanario Lisboa, Rostov, Jurong, Bojnice, Barcelona Aquarium, Loro Parque, Nikolaev, Bristol, Sea World San Diego, Cincinnati, Oklahoma City, Tulsa, Dallas Zoo, Minsk
*Memphis Zoo is listed as having 200+ mammal species and 200+ bird species, is that correct?
**Madrid Zoo is listed with 989 species, 680 of which are fish, this seems to be an error though.