As davem suggests, a list of the NASM's aircraft was a quick google away. Although the links he points to only show the displayed aircraft; for the ones socked away at Garber or loaned to other entities, you need to dig a little harder.
As to what they don't have -- well, a tremendous amount of stuff, obviously. Especially non-US stuff. You can go to the national museums of many nations, including Canada, the UK, France, Belgium, Holland, Australia, and Italy, to name a few, and see huge collections of types of which the NASM has almost none. They focus on American types, as they should.
The most significant US type that the NASM doesn't have probably is the B-24.
Their collection of US types from the 1920s and 1930s, both civil and military, is pretty thin. They are also weak on modern general aviation aircraft, ultralights, homebuilts, and rotorcraft.
In 1989, the NASM formulated a list of aircraft it lacked and of which it planned to acquire 80 over the next 30 years. Here is the list. It gives you a good idea of what NASM thinks it needs, or did 20 years ago anyway. They have acquired quite a few of these items, but still have some work to do!
Polikarpov Po-2
Douglas DC-4
De Havilland DHC-2 Beaver or DHC-3 Otter
Vickers Viscount
Embraer Bandeirante
De Havilland DHC-6 Twin Otter
Boeing 707
Boeing 727
Sud/Sud-Est/Aerospatiale SE 210 Caravelle
Tupolev Tu-104
Boeing 747
Any World War I British aircraft (e.g., Sopwith 1 1/2 Strutter)
Any World War I Italian aircraft (e.g., Ansaldo S.V.A. 9)
Any Interwar (1919-1939) British aircraft, e.g., De HavillandD.H. 82 Tiger Moth
Morane Saulnier 127
Consolidated B-24 Liberator
Douglas A-20 Havoc
Avro Lancaster
Gloster Meteor
Any German bomber (e.g., Junkers Ju 87)
Ilyushin Il-2 Shturmovik
Boeing B-52 Stratofortress
Convair F-102 Delta Dagger
Lockheed SR-71 Blackbird
Dassault Mirage III
Mikoyan-Burevich MiG-21 Fishbed
Grumman F-14 Tomcat
Alexander Eaglerock A-2, American Eagle A-1 or Brunner-WinkleBird
Travel Air 4000, Laird Speedwing, Commercial or Stearman C3B
Fleet 2 or Great Lakes 2-T-1A1
Piper Malibu
Beech Baron 55
Piper Aztec
Beech King Air
Rockwell Commander
Cessna Citation I
Stinson SB-1
Bellanca Aircruiser
Piper Pawnee or Schweitzer Grumman AG Cat
Ayres Turbo Thrush S2R-T65/400
Champion Citabria or Decathalon
Christen Eagle
Mudry CAP 10B
Akaflieg Stuttgart FS-24 Phonix
ASW 12
ASW 22B
Bekas N
Caproni A-21S
D.F.S. Olympia
D.F.S. Wiehe
Eipper-Formance Quicksilver
Glaser-Dirks DG-400
Glasflugel H-301 Libelle
Hall Cherokee II
Hoffman H-36 Dimona MK II
Letnany L-13 Blatnik
Nimbus 3/24.5
Slingsby T.21B
Woodstock One
Aerocar Micro-Imp
Aerolites Ag-Bearcat
Circa Reproductions Sopwith Triplane
Freedom Master Air Shark I
Glasair III
Hamilton HX-321
Holcomb Perigee
Lancair 200/235
Ligeti Stratos
Monnett MoniParaplane
Pietenpol Air Camper
Polen Special
Questair Venture
SA-60 Silhouette I
Sadler A22
Seawind International Seawind 2000
SNS-9 EXP II Hiperlite
Starlite SL-1
Van's RV-3
Volmer Sportsman
Wheeler Express
Bell AH-1G Cobra
Bell-Boeing V-22 Osprey
Boeing-Vertol CH-47 Chinook
Gyrodyne XRON
McDonnell Douglas AH-64A Apache
McDonnell Douglas Notar
Mil Mi-12
Mil Mi-24 Hind
Sikorsky H-3
Sikorsky HH-52
Farrington Air and Space-18A
Williams Research X-Jet
August
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