FOUR As

Some organisations using the acronym AAAA:

American Association of Advertising Agencies
Australian Automotive Aftercare Association
Alberta Aerial Applicators Association
Association of Accredited Advertising Agents
American Association of Amateur Astronomers
Association of Accountants and Auditors of Armenia
Arizona Antique Aircraft Association
American Association for Affirmative Action
Atlanta Area Aquarium Association
Acrophobia Addicts Anonymous Association
Army Aviation Association of America
Akron Area Arts Alliance
African American Appalachian Arts
Alps-Adria Acoustics Association
Albuquerque Aerostat Ascension Association
Arizona Asian American Association
American Academy of Adoption Attorneys
Association for Applied Animal Andrology
Asociación Argentina ‘Amigos de la Astronomía’

And here is an alphabet of '_a_a' words:

AAAA - a small, slender grade of battery often used in pen torches
baba - a small cake soaked in rum syrup OR a Polish coffee cake OR a Slovakian bake made of potatoes, barley and bacon
Caca - a giantess and minor goddess in Roman mythology; in one story, she betrays her cattle-rustling brother Cacus to Hercules
dada - the French word for hobby-horse, giving its name (chosen it is said, at random from a dictionary) to a radical arts movement of the early 20th century
eaea - a Polynesian name for the plant spurge; a supernatural turtle in Hawaiian folklore; a vocal styling of elongated vowels used in Hawaiian falsetto singing; a gene (eaeA) which is associated with attaching and effacing lesions and which encodes intimin, a protein
Fafa - an island, part of Tonga; a Micronesian foodstuff made from pounded taro root and coconut
gaga - prone to lightheartedness or mental instability
haha - a sunken ditch or wall built so as to stop animals trespassing but to avoid spoiling a view; originally a French term, it is usually hyphenised in English
iaia - the Catalan word for ‘granny’; also an Italian surname - Cristina Iaia is a celebrated fashion designer best known for her upmarket lingerie
Jaja - Jaja of Opobo was a king in 19th century Western Africa who was one of the first to oppose colonial trade monopolies and who was eventually deported to the W Indies for his ‘breaking of treaties’
kaka - the brown parrot of New Zealand; (with cap.) a Brazilian football star
lala - a term applied to the excesses or pretensions of Los Angeles or especially Hollywood (Lala-Land); a style of shoe made by Clarks; a fairly common female first name
mama - This child’s word for mother exists in languages as diverse as Swahili, Indonesian, Russian and Turkish and is thought to derive from a baby’s natural vocal sound when wanting to be fed; also common as a song title: songs called ‘Mama’ have been recorded by Herb Alpert, ELO, Roy Orbison, Annie Lennox, Sugarcubes, The Spice Girls, Bay City Rollers, Connie Francis, Genesis, Toto, Elvis and many more
Nana - common word for a grandmother; French slang for a young woman; the dog-maid in Disney’s Peter Pan; a novel by Zola; a type of dwarf pomegranate tree
oaoa - depending on pronunciation, this Polynesian word can mean ‘joy’, a stake to tie a canoe to, or ‘a scarecrow, made by rattling together empty water-bottles made of cocoa-nut shells’
papa - a general word meaning father; a priest of the Eastern Orthodox church; the nickname of Ernest Hemingway; a word used in the names of several islands in Orkney and Shetland, signifying that they were once inhabited by monks
qaqa - Arabic word for warrior (Bin Laden is sometimes called ‘al-Qaqa‘); Uiluit Qaqa is a mountain in Greenland; the qaqa-loro is an endangered Bolivian parrot
rara - comonly used in ‘rara avis’, a rare bird; a lake in Western Nepal; a style of musical band in Haiti
sasa - a species of woody, evergreen Asian grass; a fairly common Eastern European male first name; a port in Mindanao in the Philippines; a town in Israel
Tata - one of India’s biggest companies, best known for engineering and vehicle manufacture; a village in Senegal; a town in Morocco; a city in Hungary
uaua - a sub-race of people existing in North East Africa in the early centuries AD; Samoan for ‘to start a plantation’’; the Kiribati word for swimming; a variety of taro - piko uaua
Vava - the first player to score in two soccer World Cup finals, representing Brazil in 1958 and 1962; a township in Mozambique; ‘A Vava Inouva’ (’My Little Father‘) is a popular North African song
Wawa - a town in Ontario, Canada; a coffee shop company based in Pennsylvania; the Chinese word for ‘doll’;
Xaxa - a royal palace in the story The Mastermind of Mars by Edgar Rice Burroughs; a supernatural people in the myths of the Salish Indians of British Columbia
yaya - the Swahili word for ‘nurse’ or ‘babysitter’; also means nursemaid in Tagalog (an unrelated language); a fairly common boy’s name in W Africa
Zaza - an ethnic group from Turkey; a movie about a French entertainer, based on a play and made twice with the eponymous character played by Gloria Swanson (1923), then Claudette Colbert (1939); the cat character in children’s series Hector’s House (Le Maison de Toutou in the original French version)